Rules, unique-sum combinations, and the crossing-run logic that cracks cross-sum puzzles
Kakuro is a number-placement puzzle usually described as a crossword with digits. Instead of word clues, the dark cells of the grid carry totals, and the white cells in each horizontal or vertical run must add up to the total that heads them. The puzzle ran in American puzzle magazines under the name Cross Sums before the Japanese publisher Nikoli popularised it in Japan, where the name Kakuro - a contraction of the Japanese words for "addition" and "cross" - took hold and later travelled back to newspapers around the world.
Unlike Sudoku, Kakuro does involve arithmetic, but only light addition. The real skill is knowing which sets of digits can make a given total and combining that knowledge across runs that intersect. A handful of memorised combinations does most of the heavy lifting; the rest is careful cross-checking.
Every white cell belongs to exactly one across run and one down run, and both totals must hold at once. That double membership is what makes the puzzle solvable by deduction.
On PlayVersusAI every run is at least two cells long, and the generator builds each answer under the no-repeat rule, so every combination table below applies exactly. The checker enforces the same rule: the puzzle completes only when every run hits its total with no digit used twice.
Most totals can be split several ways, but some have only one set of distinct digits. These are the entry points into every puzzle:
The pattern is easy to remember: the smallest and largest possible totals for each run length, and the totals one step in from each end, have a single combination. A combination tells you which digits are in the run, not their order - working out the order is what crossing runs are for.
Totals in the middle of the range have several combinations, but they still exclude digits. A 5 across two cells is 1+4 or 2+3, so it can never contain 5 or anything higher. A 15 across two cells is 6+9 or 7+8, so nothing below 6 fits. A 12 across two cells is 3+9, 4+8 or 5+7; the digits 1 and 2 are impossible, and 6 only works as 6+6. The board has no pencil-mark mode, so jot these ranges on paper for the cells you are working on.
This is the single most powerful idea in Kakuro: a cell's digit must belong to both its across combination and its down combination. Take an across run of two cells totalling 16, which must hold 7 and 9. Its left cell also starts a down run of three cells totalling 23, which must be 6, 8 and 9. The only digit in both sets is 9, so the shared cell is 9, the other across cell is 7, and the remaining two cells of the down run hold 6 and 8 in some order. Now check what crosses those two cells. If one of them sits in an across run of two totalling 7 (1+6, 2+5 or 3+4), it cannot be 8, so it is 6 and the last cell is 8. Three runs, four cells, all resolved without a guess.
Small totals cross just as neatly. Where a two-cell run totalling 3 (1 and 2) meets a two-cell run totalling 4 (1 and 3), the shared cell can only be 1, which immediately fixes both partners.
When all but one cell of a run is filled, the last cell is the total minus the rest. The same trick works part-way through: if a four-cell run totals 20 and two of its cells already hold 9 and 8, the remaining pair must make 3, which can only be 1 and 2. Turning a long, ambiguous run into a short remainder often converts it into a unique-sum case.
Two-cell runs have the fewest combinations of all, and on PlayVersusAI grids they cluster around the extra clue blocks scattered through the interior. Scan for two-cell runs with an extreme total - 3, 4, 16 or 17 - before anything else. Each one pins down two digits as a pair, and every crossing run it touches inherits that constraint.
The three buttons above the board choose the grid size. The size counts the clue row along the top and the clue column down the left, and interior cells can become extra clue blocks, so the number of white cells varies from puzzle to puzzle. Any cell that would leave a run of a single cell is turned into a clue block as well.
| Grid | Playing area | What to expect |
|---|---|---|
| 6×6 | 5×5, roughly 15-20 white cells | Short runs and small totals. Ideal for learning the clue layout and the two-cell combinations. |
| 8×8 | 7×7, roughly 30-40 white cells | The default. Runs up to seven cells long and many more crossings to keep track of. |
| 10×10 | 9×9, roughly 50-65 white cells | Long runs with large totals, where the middle-range combinations and crossing logic do most of the work. |
The generation method is the same at every size, so difficulty grows with scale rather than with a different style of logic. A move is one digit typed, so the minimum is the number of white cells. Your best is stored in this browser and, when you are signed in, submitted to the leaderboard, where all three sizes share one fewest-moves table.
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