Kakuro: the crossword of numbers

Updated 2026-10-30

Kakuro is a number puzzle laid out like a crossword: you fill white cells with digits 1 to 9 so that each horizontal and vertical run adds up to its clue, with no digit repeated within a run. It mixes simple addition with Sudoku-style logic, and every good Kakuro has exactly one solution.

Kakuro rules

A Kakuro grid contains black cells and white cells. Some black cells are split diagonally and hold one or two numbers, the clues. A clue in the upper-right half refers to the run of white cells to its right (an across run); a clue in the lower-left half refers to the run of white cells below it (a down run). The rules are:

• Write a single digit from 1 to 9 in every white cell.
• The digits in each run must add up to the run's clue.
• A digit may not appear twice in the same run.

That is all. There are no rows, columns or boxes in the Sudoku sense; a digit can appear many times in the grid as long as it does not repeat within one run. Kakuro is also known in some places as Cross Sums.

Unique combinations: the key to Kakuro

Because digits cannot repeat and only 1 to 9 are allowed, some clues can be made in exactly one way. Learning these is the biggest single step from beginner to confident solver.

The table tells you which digits go in the run, not their order. Placing them is where the logic comes in.

Useful non-unique facts are worth knowing too. A two-cell run with an even clue can never contain half that clue: a 10 in two cells is 1+9, 2+8, 3+7 or 4+6, never 5+5. And a two-cell 5 is either 1+4 or 2+3, so it can never contain a 5 or anything larger.

How to solve Kakuro: strategies

1. Start at the intersections of unique runs

Every white cell belongs to one across run and one down run, so its digit must suit both. Suppose a cell is part of an across 3 (digits 1 and 2) and a down 4 (digits 1 and 3). The only digit the two sets share is 1, so the cell is 1. Similarly, where a two-cell 16 (7, 9) crosses a two-cell 17 (8, 9), the shared cell must be 9.

2. Use upper and lower limits

Even without a unique combination, you can bound a cell. In a two-cell run of 6, the options are 1+5 and 2+4, so neither cell can be 3 or more than 5. In a three-cell run of 22, the other two cells add up to at most 9+8 = 17, so the third must be at least 5.

3. Eliminate by crossing runs

Write small candidates in uncertain cells, just as in Sudoku. Each time you narrow one run, check the runs that cross it. A cell that can only be 8 or 9 from its across run, crossed with a down run that excludes 9, is solved.

4. Finish runs by subtraction

Once all but one cell of a run is filled, the last cell is simply the clue minus the sum of the others. This sounds trivial, but late in a puzzle these quick finishes often cascade across the grid.

5. Look at the whole block

Sometimes a compact area of white cells is fully described by a set of across clues and also by a set of down clues. Both sets must add up to the same total. If one cell sticks out of the block, the difference between the totals gives its value, similar to the 45 rule in Killer Sudoku.

Common mistakes

• Repeating a digit in a run because the sum works out, such as 4+4 for 8. Not allowed.
• Applying Sudoku rules where they don't belong. Two cells in the same row but different runs may share a digit.
• Reading a clue for the wrong direction. The upper-right number is across, the lower-left number is down.

If you like Kakuro

Kakuro rewards the same habits as many logic puzzles: find the most constrained spot, make one sure deduction, and let it ripple outward. If you want that feeling without any arithmetic, Star Battle is a good next step. You place stars so each row, column and region gets its quota, and no two stars touch. Learn the Star Battle rules, or explore more puzzles like Sudoku.