Cross-reference

Cross-reference is the kakuro reasoning at the intersection of two runs. Every white entry cell belongs to exactly two runs, one horizontal and one vertical. Whatever digit lives in the cell must be valid in both runs — appearing in the horizontal run's combination and the vertical run's combination. The intersection of the two runs' digit sets is the cell's possible digit set, and it's usually small.

The basic move

Suppose a cell sits at the intersection of a 3-cell horizontal run with target 6 and a 2-cell vertical run with target 4. The horizontal run is a unique sum forcing {1, 2, 3}. The vertical run is a unique sum forcing {1, 3}. The cell must be a digit appearing in both sets — the intersection of {1, 2, 3} and {1, 3} is {1, 3}. So the cell is either 1 or 3.

Now apply the same reasoning to the other cells of each run. The cells will produce intersections of their own; together, the constraints often narrow the cells further. If one of the horizontal run's other cells also has {1, 3} as its possible digit set, kakuro-pair reasoning fires across the run.

Why it's the kakuro workhorse

Kakuro reasoning is fundamentally about combining run-level information with cell-level information, and cross-reference is the move that does that combining. Every kakuro placement that isn't directly forced by a unique sum is forced by some cross-reference: the cell's two runs together pin the digit.

A practical rhythm for kakuro solving: scan for unique sums first, then run cross-reference at every intersection that involves an already-forced run. Most cells in most kakuros place in this two-pass process. The cells that don't are where the more advanced techniques — kakuro-pair, forced-cell-kakuro, kakuro-triple — start to matter.

A note on partial cross-reference

Cross-reference often fires before either run is fully forced. A 4-cell run with target 22 has multiple valid combinations, but if the cells' other runs have narrowed enough that only one combination intersects consistently with all of them, the move still fires. The bookkeeping is more demanding — the player tracks several possible combinations for the run and prunes them as cross-references rule them out — but the underlying principle is the same intersection logic.

This partial-information variant is the kakuro analogue of partial-cage-combinations in killer Sudoku: tracking what's still possible as constraints accumulate is the core skill.