Kakuro Cross-Sum Solver
A number crossword — fill white cells with digits 1–9 so each horizontal and vertical run sums to the clue, with no repeated digits in any run.
About Kakuro
Kakuro (クロスサム, "cross-sum") is a logic puzzle that blends crossword structure with arithmetic constraints. The grid contains white answer cells separated by black cells. Clue cells display a diagonal number: the upper-right value is the required sum for the cells running across to the right; the lower-left value is the required sum for the cells running down. Each run of white cells must use unique digits from 1 to 9.
Unlike Sudoku, there is no requirement that all digits appear in every row or column — only that each run's digits are unique and sum correctly. Kakuro puzzles always have a unique solution derivable by pure logic. The solver uses constraint propagation to find valid digit combinations for each run, then resolves conflicts with backtracking.
Frequently Asked Questions
What is a "run" in Kakuro?
A run is a consecutive sequence of white cells in a row (across) or column (down), bounded by black or clue cells. Each run has a target sum shown in the adjacent clue cell.
Can digits repeat in a run?
No. Each digit 1–9 may appear at most once within a single run. This is what makes Kakuro challenging — possible digit combinations for a given sum are limited.
How do I read the clue cells?
A clue cell is split diagonally. The upper-right number is the across clue (sum of cells to the right). The lower-left number is the down clue (sum of cells below).
How does the Kakuro solver work?
The solver first computes all valid digit combinations for every run (subsets of {1–9} with the right length and sum). It then intersects the possible values for each cell, and uses backtracking to resolve remaining ambiguities.
How to Play Kakuro
1 Rules
- ✓ Fill white cells with digits 1–9.
- ✓ Digits in each horizontal run must sum to the across clue.
- ✓ Digits in each vertical run must sum to the down clue.
- ✓ No digit repeats within a single run.
2 Strategies
- 💡 Unique Combos: A 2-cell run summing to 3 can only be {1,2}. A 2-cell run summing to 16 can only be {7,9}.
- 💡 Intersection: Cells shared by across and down runs must satisfy both constraints simultaneously.
- 💡 Min/Max: The minimum sum for k cells is 1+2+…+k. The maximum is (9-k+1)+…+9.
Reading clues: The upper-right number in a clue cell is the across sum; the lower-left number is the down sum. Click a white cell and type a digit, or use the keyboard.