3 in 2 cells
{1, 2}
Only one valid combination — both digits are forced. Crossings decide which cell gets which digit.
Official rules
Kakuro Rules
Kakuro is a number logic puzzle played on a crossword-style grid. Each run of white cells must sum to its clue using digits 1–9, with no digit repeated in any single run. The puzzle has exactly one solution, always reachable by deduction alone.
The five core rules
- 1
Fill white cells only.
Black cells are fixed — they contain clue numbers or are structural dividers. Only the white cells receive digits.
- 2
Use digits 1 through 9.
Each white cell gets exactly one digit from 1 to 9. Zero is never used in Kakuro.
- 3
Each run must sum to its clue.
A run is a consecutive sequence of white cells (horizontal or vertical) preceded by a clue cell. The digits in that run must add up exactly to the clue number.
- 4
No digit repeats within a run.
Within any single run, all digits must be different. For example, a 3-cell run summing to 6 cannot use {2, 2, 2} — it must use distinct digits like {1, 2, 3}.
- 5
Logic, not guessing.
Every puzzle on Free Kakuro has exactly one valid solution. If you find yourself guessing, backtrack and look for a forced move you missed.
Understanding the Kakuro grid
A Kakuro grid contains two types of cells:
- White cells — the cells you fill with digits. Each white cell belongs to exactly two runs: one across (horizontal) and one down (vertical).
- Black cells — either solid dividers or clue cells. A clue cell is split diagonally: the number in the upper-right triangle is the across-sum for the horizontal run starting immediately to its right; the number in the lower-left triangle is the down-sum for the vertical run starting immediately below it.
A run is a maximal consecutive sequence of white cells in the same row or column bounded by black cells or the grid edge. Runs can range from 2 cells (minimum) to the full width or height of the grid.
How sum clues work
Every run has a clue — a target sum the digits in that run must equal. The length of a run limits which digit combinations are possible. Short runs with low or high sums are often forced to a single combination, giving you a guaranteed starting point.
16 in 2 cells
{7, 9}
Another forced pair. High 2-cell sums narrow just as quickly as low ones.
6 in 3 cells
{1, 2, 3}
One combination, three cells. Place the set and let intersecting runs determine the order.
10 in 3 cells
{1, 2, 7} or {1, 3, 6} or {2, 3, 5} or {1, 4, 5}
Four combinations — not forced. Use cross-run intersection to narrow the candidates.
For a complete list of every valid digit set for every sum and run length, use the Kakuro combinations reference.
How to win
The puzzle is complete when every white cell contains a digit from 1 to 9, every run sums exactly to its clue, and no digit is repeated within any run. There is exactly one arrangement of digits that satisfies all constraints simultaneously — and finding it through pure deduction is the point of the game.
You never need to guess. If the puzzle feels stuck, look for a run you have not fully analyzed. Check for forced combinations (runs where only one valid digit set exists), then use the cross-run intersection technique to eliminate candidates in shared cells.
Applying the rules: a step-by-step example
- 1
Identify forced runs. Find any run where only one digit combination is numerically valid. A 2-cell run summing to 3 must be {1, 2}. Place those candidates immediately.
- 2
Check intersecting runs. Each white cell belongs to both an across run and a down run. If a digit is not a candidate in one run's valid combinations, remove it from the crossing cell.
- 3
Place confirmed digits. When a cell has only one candidate remaining after intersection, place it. This often forces adjacent cells and cascades through the grid.
- 4
Use residual sums. Once some cells in a run are filled, subtract those digits from the clue. The remaining sum and the remaining empty cells form a shorter sub-problem, often with fewer valid combinations.
- 5
Repeat until solved. Keep cycling through these steps. Every placement narrows the search space. A valid puzzle always resolves without guessing.
The full rules guide walks through a complete worked example with an annotated grid. For in-depth technique coverage, visit the technique library.
Kakuro rules FAQ
- Can a digit appear more than once in the same run?
- No. Within any single run, each digit 1–9 can appear at most once. The same digit may appear in other runs elsewhere on the grid — the no-repeat rule only applies within each individual run.
- Can I use 0 in Kakuro?
- No. Only digits 1 through 9 are used in Kakuro. Zero is never valid in any cell.
- What is the minimum run length in Kakuro?
- Two cells. A single white cell with no adjacent white cells in the same direction cannot form a valid run, so Kakuro grids always ensure runs are at least 2 cells long. This guarantees meaningful sum constraints.
- How many solutions does a Kakuro puzzle have?
- Exactly one. A well-formed Kakuro puzzle has a unique solution. Every puzzle on Free Kakuro is generated with a uniqueness check — any grid that admits multiple solutions is discarded and regenerated.
- Is Kakuro the same as Cross Sums?
- Yes. Cross Sums is the older American name for the same puzzle. Kakuro (カックロ) is the Japanese name that became dominant internationally. Both names refer to the same rules and grid structure. Learn more at Kakuro cross sums.
Ready to apply the rules?
Free Kakuro generates unlimited puzzles at four difficulty levels, all in your browser with no sign-up. Start with easy puzzles to practice the core rules, then work up to harder grids that demand advanced techniques. The beginner guide is a good first stop if you are new to the puzzle.