Unique sum
A kakuro run whose cell count and target sum together force exactly one valid digit combination — the most direct deductive move in the puzzle.
A unique sum is a kakuro run whose cell count and target sum together force exactly one valid digit combination. The combination's digits are determined; only their arrangement within the run's cells remains to be solved. Unique sums are the most direct deductive move in kakuro and the first thing most solvers scan for on any new grid.
The standard unique sums
The unique-sum entries from the kakuro sum-combinations table are worth memorising by sight:
For 2 cells: target 3 forces {1, 2}. Target 4 forces {1, 3}. Target 16 forces {7, 9}. Target 17 forces {8, 9}.
For 3 cells: target 6 forces {1, 2, 3}. Target 7 forces {1, 2, 4}. Target 23 forces {6, 8, 9}. Target 24 forces {7, 8, 9}.
For 4 cells: target 10 forces {1, 2, 3, 4}. Target 11 forces {1, 2, 3, 5}. Target 29 forces {5, 7, 8, 9}. Target 30 forces {6, 7, 8, 9}.
For 5 cells: target 15 forces {1, 2, 3, 4, 5}. Target 35 forces {5, 6, 7, 8, 9}.
The pattern is the extremes — the smallest possible target for a given cell count is forced; the largest is forced; and the next one in from each extreme is also forced.
Why these are the kakuro foundation
A unique-sum run pins every cell in the run to one of a small set of digits — the digits in the forced combination. That immediately narrows every cell's candidates and feeds into the cross-reference move at every intersection. A 3-cell run forced to {1, 2, 3} provides three cells where the digits 1, 2, and 3 must each go; if one of those cells also belongs to a column run constrained to {2, 5, 7}, the cell must be 2, and the puzzle gets one digit closer to solved.
Most kakuros are designed so that several unique sums fire on the opening pass. Identifying them is the kakuro equivalent of the first-pass naked-and-hidden-singles scan in Sudoku.
After the unique-sum scan
Once every unique-sum run has been pinned to its forced combination, the harder reasoning takes over: cross-reference for cells, partial-cage-combinations-style narrowing for runs as cells fill, and kakuro-pair / kakuro-triple for unit-spanning eliminations. The unique-sum scan opens the puzzle; the other techniques close it.
See also
- Kakuro sum combinations— The reference table mapping each (cell-count, target-sum) pair to its valid digit combinations. The foundational lookup that anchors most kakuro deduction.
- Run (kakuro)— A horizontal or vertical sum-segment in a kakuro grid. The white cells from one clue cell to the next, which must sum to the clue's target without repeating a digit.
- Kakuro— A Japanese number-placement puzzle. Cross-sums act like a crossword grid; players fill white cells with digits 1–9 so each run sums to its target without repeating a digit.
Read more
- Meet Kakuro
An introduction to Kakuro — what it is, how it differs from Sudoku, and why people who like number-logic puzzles often end up preferring it.