Forced cell (kakuro)

A forced cell in kakuro is a placement determined by the intersection of cross-reference and run constraints. The cell's two runs, taken together with the digits already placed elsewhere, leave only one possible digit for the cell. It's the kakuro equivalent of a hidden single in Sudoku — a placement where every alternative has been ruled out by combining constraints.

How forced cells emerge

Two routes produce forced cells. The first is direct: a cell's cross-reference set narrows to one digit. If a cell's horizontal run admits {1, 4, 6} and its vertical run admits {2, 4, 7}, the cross-reference is {4} alone, and the cell is forced to 4. This route is the most common forced-cell scenario early in a puzzle.

The second route is digit-first within a run. A run admits some combination, say {1, 3, 5, 7}, and one of those digits — say 7 — has only one cell in the run that could hold it (the other cells, through their other runs' constraints, can't be 7). The 7 is forced into that cell. This is the kakuro equivalent of a hidden single's logic: the digit's possible cells in the unit narrow to one, even though the cell itself might still have multiple cross-reference candidates.

The two routes complement each other. Cross-reference forced cells emerge cell-first; digit-first forced cells emerge run-first. Most kakuro solvers naturally alternate between the two perspectives as the puzzle resolves.

Why it earns its own glossary entry

In Sudoku, the parallel concepts split between naked single (cell-first, "this cell has only one candidate") and hidden single (digit-first, "this digit has only one candidate cell"). In kakuro, the two routes blur because cross-reference is itself a hybrid — it's cell-first in form (narrow this cell's possible set) but it uses run-level information (which digits the run admits). The forced-cell label covers both ways the move can fire.

A practical solving rhythm: after every placement, re-run cross-reference on every cell that shares a run with the placed digit, and check whether any cell's set has narrowed to a single digit. If yes, place that digit too and repeat. Forced-cell cascades — one placement triggering several more — are common on well-designed kakuros and form the bulk of the late-game solving.

When you'll see it

Constantly, on every kakuro past easy. Forced cells are the closing move on most runs: once cross-reference and unique-sum have narrowed the run's combinations and a few cells have been placed, the rest typically force one at a time. The skill is in maintaining the cross-reference sets cleanly as placements happen — sloppy bookkeeping causes solvers to miss forced cells that should have been obvious.