Kakuro Puzzles
A crossword shape filled with digits instead of letters. Every run of white cells must add up to the clue printed before it, using the digits 1 to 9, and no digit may repeat inside a run. Stuck? The board replays one deduction at a time and tells you what forced it.
Also published as cross sums, and as kakkuro in Japan, where Nikoli named it from a contraction meaning addition cross.
What Is Kakuro?
Kakuro is a logic puzzle built on a crossword grid, but the answers are numbers rather than words. Each unbroken line of white cells — a run — carries a total in the black cell before it, and you fill the run with digits from 1 to 9 that add to that total. The one extra rule does all the work: no digit may repeat inside a run. Nikoli, the Japanese publisher that also popularised sudoku, gave it the name in 1986 as a contraction of kasan kurosu, addition cross.
A black cell can carry two totals, one for the run heading right and one for the run heading down, which is why those cells are split by a diagonal. The number above the line belongs to the across run; the number below it belongs to the down run.
The Rule That Trips Up Sudoku Players
In sudoku a digit appears once per row. In kakuro it appears once per run, and those are not the same thing. If a black cell breaks a row into two runs, the digit 7 can sit in both halves quite legally — they are different runs, and nothing connects them.
Count the run, not the row. Ruling out a digit because it already appears elsewhere in the same row is the commonest way to convince yourself a perfectly good kakuro is broken. The only line that matters is the unbroken stretch of white cells between two blocks.
The Runs That Name Themselves
Most clue totals can be made several ways. A few can be made exactly one way, and those are where every kakuro is opened. They are not a trick to memorise so much as an observation about how little room the digits 1 to 9 leave at the extremes: to reach the smallest possible total you must use the smallest digits, and there is only one way to do that.
| Cells in the run | Lowest two totals | Highest two totals |
|---|---|---|
| 2 | 3 = 1+2 · 4 = 1+3 | 16 = 7+9 · 17 = 8+9 |
| 3 | 6 = 1+2+3 · 7 = 1+2+4 | 23 = 6+8+9 · 24 = 7+8+9 |
| 4 | 10 = 1+2+3+4 · 11 = 1+2+3+5 | 29 = 5+7+8+9 · 30 = 6+7+8+9 |
There are exactly four at every run length — two at the bottom of the range and two at the top — and the pattern continues for longer runs. Many kakuro guides quote only 3 and 17; the full set is worth knowing, because 4 and 16 are just as forced and turn up just as often.
Crossing A Low Run Against A High One
The single most productive move in kakuro is not arithmetic, it is comparison. Every white cell belongs to two runs at once — one across, one down — so whatever digit it holds has to be legal in both. Find a cell where a run forced low crosses a run forced high and the two sets of candidates often share exactly one digit.
A three-cell run totalling 7 must be 1, 2 and 4. A two-cell run totalling 17 must be 8 and 9. Where those cross, there is nothing in common at all — which tells you the layout is impossible, so you have misread a clue. Where a 7-in-three crosses a 16-in-two, only the digit 7 appears in both, and the cell is settled without adding anything up.
Why These Boards Are Stored Rather Than Generated
Every other grid puzzle on this site is built fresh when the page loads. Kakuro is the exception, and the reason is worth stating plainly rather than hiding behind a loading spinner.
A kakuro cell is held by only two constraints — its across run and its down run. A killer sudoku cell is held by four: its row, its column, its box and its cage. Two is not enough to force a single answer, so a randomly filled kakuro almost always has several solutions. Measured here across millions of arrangements, roughly one in three thousand comes out uniquely solvable.
| Puzzle | Constraints on one cell | Random arrangement is unique | Built when |
|---|---|---|---|
| Killer sudoku | 4 — row, column, box, cage | Almost always, with givens | Page load |
| Calcudoku | 3 — row, column, cage | Often | Page load |
| Futoshiki | 3 — row, column, signs | Often | Page load |
| Kakuro | 2 — across run, down run | About 1 in 3,000 | Found offline, stored |
So the winners were found by a long offline search and stored, which is much closer to how kakuro is actually published: setters design and test layouts rather than roll them fresh each morning. The consequence is an honest one — the inventory here is sixty boards, not unlimited. Every one of them earned its place by passing two separate machine checks, described below.
Proved To Have One Answer, And To Need No Guess
A board is only kept if both of these hold, and they are different questions:
- Exactly one solution. An exhaustive search counts the ways the grid can be completed and stops at two. A board with two answers is not a puzzle, and it is thrown away.
- Solvable by reasoning alone. A separate solver that only ever uses techniques a person could use — forced combinations, crossings, the no-repeat rule — has to finish the whole grid unaided. If it stalls and would need a guess, the board is thrown away too.
That second check is what Show me the next move replays. It gives you one deduction, names the cell and says what forced it, then stops. It is not a solve button: the answer stays yours to find.
Why A Kakuro Here Is A Small One
The boards run from 16 to 24 white cells. That is a small kakuro — some sites offer grids up to 30 squares across, and if you want a long evening's puzzle those are the better choice. The trade is deliberate: the uniqueness rate collapses as grids grow, and past roughly 24 cells an offline search stops finding boards that are both uniquely solvable and free of guesswork. Rather than relax either guarantee for the sake of a bigger grid, the grids stay small and the guarantees stay true.
Kakuro Puzzles Frequently Asked Questions
Can a number repeat in kakuro?
Not inside one run. It may appear more than once in the same row of the grid, provided a black cell separates the two, because that makes them different runs. Judging a digit by its row rather than its run is the usual reason a good puzzle looks wrong.
Is zero ever used?
No — only 1 to 9. That is why a two-cell run can never total less than 3 or more than 17, and why the forced combinations exist at all.
What are the magic numbers in kakuro?
The totals that allow only one set of digits. At two cells: 3, 4, 16 and 17. At three: 6, 7, 23 and 24. At four: 10, 11, 29 and 30. There are exactly four at every length — the table above has them.
How is kakuro different from sudoku?
Sudoku gives you a fixed 9×9 and asks you to place nine of each digit. Kakuro gives you a crossword shape and asks you to hit a total on every line. The arithmetic is the visible difference; the real one is that kakuro's no-repeat rule applies to short runs rather than to whole rows.
Why only sixty puzzles?
Because a uniquely solvable kakuro is rare enough that it cannot be built while the page loads — roughly one arrangement in three thousand qualifies. The boards were found offline and stored, so the inventory is finite. Saying "sixty" is more useful than claiming a number that is not true.
Do I need an account?
No. There is no login, nothing is uploaded, and no adverts are served. Your progress stays in your own browser.
What is kakuro?
Kakuro is a logic puzzle on a crossword-shaped grid. Instead of words you enter the digits 1 to 9, and instead of clues each run of white cells carries a total it must add up to. No digit may repeat inside a single run. It is also published as cross sums, and in Japan as kakkuro.
How many kakuro puzzles are on this page?
Sixty. Unlike the other grid puzzles here, kakuro boards are stored rather than generated when the page loads, because a uniquely solvable kakuro turns up in roughly one random arrangement in three thousand. The inventory is finite and stated rather than dressed up as unlimited.
Does every kakuro here have one answer?
Yes, and it is checked rather than claimed. Every board was kept only after a search confirmed exactly one solution exists and a logical solver finished it without a single guess. A puzzle that needed a guess was thrown away.