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.
Kakuro is a Japanese number-placement puzzle that looks like a crossword grid filled with arithmetic. The grid contains white cells, where the player writes digits 1 through 9, and dark clue cells containing target sums for the horizontal and vertical runs extending from them. Each run's white cells must sum to the clue's target, and no digit may repeat within a single run. There's no global "9 cells, one of each digit" constraint like in Sudoku — the rules live entirely within each run.
How a kakuro plays
A typical clue cell shows two numbers separated by a slash — the left or top number is the column run's target sum, the right or bottom number is the row run's target. White cells fill in below or to the right. The player works the grid by combining sum constraints — "this 3-cell run must sum to 6, so it's some permutation of 1+2+3" — with the no-repeats-in-run rule and the cross-reference of intersecting runs.
The puzzle's difficulty scales with the size and the proportion of cells that admit unique combinations. A small kakuro might have only short runs of 2-4 cells; large kakuros have runs up to 9 cells, which is the most a single run can hold (one of each digit). Some grids include a 45-cell or longer compound region for advanced variants, though Sudoku Mountain's kakuros stay within standard sizes.
How it differs from Sudoku
The two puzzles share a no-repeats-in-row-or-column flavour and a deductive style, but the differences are substantial. Sudoku has a 9×9 grid with a 3×3-box constraint and one of each digit per row, column, and box. Kakuro has irregular grids where runs vary in length, no boxes, no global completeness — only the local sum and no-repeats constraints on each run.
The reasoning style differs too. Sudoku reasoning is mostly about placement: where can this digit go in this unit? Kakuro reasoning leans on combination: which sets of digits can sum to this run's target? The technique catalog includes unique sums where the combination is forced by the cell count alone, cross-reference deduction at the intersections of runs, and the kakuro-pair and kakuro-triple variants of the naked-subset techniques.
Where to start
The reference table that anchors most kakuro solving is the sum combinations lookup — for any (cell-count, target-sum) pair, what digit combinations can produce that sum? Memorising the unique-sum entries (sums with exactly one valid combination) is the kakuro equivalent of internalising Sudoku's basic techniques. After that, the rest of the technique stack opens up.
For an introduction with worked examples, see Meet kakuro.
See also
- 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.
- Clue cell— The dark cell in a kakuro grid carrying the target sums for the runs extending from it — usually one or two numbers separated by a slash.
- Entry cell— A white cell in a kakuro grid that the player fills with a digit. Each entry cell belongs to exactly two runs — one horizontal, one vertical — and must satisfy both.
- 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.
- Cross-reference— Kakuro deduction at the intersection of two runs. The cell shared between a row run and a column run must hold a digit valid in both — usually pinning the cell directly.
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.