Kakuro Solver

Kakuro is a number puzzle that combines crossword structure with arithmetic constraints. The grid has diagonal clue cells containing numbers — the number above a slash is the sum for the vertical run of cells below it, and the number below the slash is the sum for the horizontal run of cells to its right. You fill each white cell with a digit from 1 to 9 so that each run's digits add up to its clue number, with no digit repeated within any single run. Unlike Sudoku, Kakuro requires actual arithmetic, and certain sum-and-length combinations have only one possible valid set of digits — memorizing these "forced combinations" (like a sum of 3 in 2 cells can only be 1+2) is the key to solving efficiently. For each run in your puzzle, enter the target sum and the number of cells in that run, then hit Solve to see every valid digit combination for that constraint.

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About Kakuro

Kakuro (also written Ka-Kuro, from the Japanese "kasan kurosu" meaning "addition cross") is a numeric crossword puzzle that combines elements of crosswords and arithmetic in a uniquely elegant format. It was created in the United States by Dell Magazines in 1966 under the name "Cross Sums" and published regularly in Dell Pencil Puzzles & Word Games. Japanese publisher Nikoli — the same company that popularized Sudoku — discovered Cross Sums in 1984, renamed it Kakuro, and refined the puzzle format, growing it into a major puzzle brand in Japan before it spread internationally.

The puzzle grid resembles a crossword but uses numbers instead of letters. Black diagonal clue cells contain numbers — the number above a slash indicates the sum for the vertical run of white cells below it, and the number below the slash indicates the sum for the horizontal run of white cells to its right. Players fill each white cell with a digit from 1 to 9 such that: each horizontal run's digits sum to its clue, each vertical run's digits sum to its clue, and no digit repeats within any single run. The combination of arithmetic and no-repeat constraints creates a rich deductive environment.

Kakuro gained major international attention in the mid-2000s alongside the Sudoku boom, appearing in newspapers, puzzle books, and apps worldwide. The puzzle is sometimes described as "mathematical Scrabble" — it requires number placement skills (like Sudoku) and arithmetic skills (unlike Sudoku), combined with the crossword's intersecting constraint structure. Difficulty ranges from straightforward puzzles solvable by run length analysis to expert puzzles requiring multi-step deductive chains across multiple intersecting runs.

PuzzleUnlock's Kakuro solver approaches the puzzle analytically: for any combination of sum and run length, it calculates all valid digit sets (combinations that sum to the target using distinct digits from 1-9). Entering multiple intersecting runs allows cross-referencing to find digits that must appear in a given cell.

Certain sum-length combinations have exactly one valid digit set — memorizing these provides immediate placements. Sum of 3 in 2 cells: 1,2'} only. Sum of 4 in 2 cells: 1,3'} only. Sum of 16 in 2 cells: 7,9'} only. Sum of 17 in 2 cells: 8,9'} only. Sum of 45 in 9 cells: 1,2,3,4,5,6,7,8,9'} — the complete set. These forced combinations appear constantly and provide the fastest starting placements.

Cells at the intersection of a horizontal and vertical run must simultaneously satisfy both runs' sum and no-repeat constraints. These double-constrained cells are the most restricted in the puzzle. Find cells where both runs have forced or near-forced digit sets and look for the intersection of those sets — often producing a single valid digit.

For runs with only one or two unknown cells, calculate what remaining digits must sum to by subtracting known digits from the run total. A run of sum 20 with confirmed digits 3 and 8 needs its remaining cells to sum to 9 — now apply the forced-combination table to the remaining run length for that partial sum.

PuzzleUnlock's solver calculates valid digit sets for each run separately. When you enter multiple intersecting runs, the solver shows digits that must appear across all your constraints. Digits appearing in every valid combination for both the horizontal and vertical run through a cell are certain placements for that cell.

Memorize forced digit combinations

Intersection cells are the highest-value starting points

Use the complement approach for near-complete runs

Multiple runs narrow cell options multiplicatively

Q: Can I use 0 in Kakuro cells?

No — only digits 1 through 9 are valid. Zero is never used in Kakuro.

Q: What is the maximum possible sum for a single Kakuro run?

A run of 9 cells can sum to maximum 45 (using digits 1+2+3+4+5+6+7+8+9). For any shorter run of length n, the maximum sum is the sum of the n largest available digits.

Q: How is Kakuro different from Sudoku?

Sudoku has a fixed 9×9 grid with 3×3 box constraints and no arithmetic. Kakuro has a variable crossword-style grid with arithmetic sum constraints on variable-length runs. Sudoku requires logical placement only; Kakuro requires arithmetic and logical placement together.

Q: Why does the solver show multiple valid digit combinations?

Most sum-length combinations have multiple valid digit sets. The solver shows all mathematically valid options — you need crossing run information to determine which specific combination applies. Entering intersecting runs in the solver narrows options dramatically.

Q: Is there a standard Kakuro grid size?

Unlike Sudoku's fixed 9×9, Kakuro grids vary in size and shape. Common published sizes range from small 5×5 puzzles to large 16×16 grids. The puzzle's difficulty depends on grid size, average run length, and the specific combination of sum constraints.