Kakuro Puzzle and Combination Table
A third of all Kakuro clues can be made in exactly one way, and they sit at the extremes. Play a puzzle, or use the table.
A third of every clue is already solved
There are 129 distinct clues in Kakuro — every combination of run length and sum that can be made from distinct digits 1 to 9 — and 43 of them, 33 per cent, can be made in exactly one way. A three-cell run summing to 6 is 1+2+3 and nothing else. A two-cell run summing to 17 is 8+9. You do not deduce those, you recognise them, and that recognition is where every solve starts.
The forced ones are not scattered about. At every run length from two to seven there are exactly four of them — the lowest sum and the next one up, the highest and the next one down. The reason is short: the smallest run of k digits is 1+2+…+k, and the only way to make one more is to push the top digit up by one; the same argument runs downward from the top. Two more than the minimum is the first sum you can make two ways, so the forced band is exactly two deep at each end.
That is what makes the table below worth learning rather than looking up. The useful part of it is small and structured — four clues per length, at the extremes — not 129 rows to be searched one at a time.
The forced clues, all 43 of them
These have one combination each. Find them first.
| Cells | Range | Forced sums and their digits |
|---|---|---|
| 2 | 3–17 | 3 = 1+2 · 4 = 1+3 · 16 = 7+9 · 17 = 8+9 |
| 3 | 6–24 | 6 = 1+2+3 · 7 = 1+2+4 · 23 = 6+8+9 · 24 = 7+8+9 |
| 4 | 10–30 | 10 = 1+2+3+4 · 11 = 1+2+3+5 · 29 = 5+7+8+9 · 30 = 6+7+8+9 |
| 5 | 15–35 | 15 = 1+2+3+4+5 · 16 = 1+2+3+4+6 · 34 = 4+6+7+8+9 · 35 = 5+6+7+8+9 |
| 6 | 21–39 | 21 = 1+2+3+4+5+6 · 22 = 1+2+3+4+5+7 · 38 = 3+5+6+7+8+9 · 39 = 4+5+6+7+8+9 |
| 7 | 28–42 | 28 = 1+2+3+4+5+6+7 · 29 = 1+2+3+4+5+6+8 · 41 = 2+4+5+6+7+8+9 · 42 = 3+4+5+6+7+8+9 |
One-cell and eight-cell runs are all forced for a different reason — there is only one digit to place, or only one left out — and the single nine-cell run is 45. Those are left off the table above because they follow from the rules rather than needing to be remembered.
Every clue, and how many ways it can be made
Enumerated from all 511 subsets of 1 to 9, not typed out.
| Cells | Sum | Ways | Combinations |
|---|---|---|---|
| 2 | 3 | 1 | 12 |
| 2 | 4 | 1 | 13 |
| 2 | 5 | 2 | 23 14 |
| 2 | 6 | 2 | 24 15 |
| 2 | 7 | 3 | 34 25 16 |
| 2 | 8 | 3 | 35 26 17 |
| 2 | 9 | 4 | 45 36 27 18 |
| 2 | 10 | 4 | 46 37 28 19 |
| 2 | 11 | 4 | 56 47 38 29 |
| 2 | 12 | 3 | 57 48 39 |
| 2 | 13 | 3 | 67 58 49 |
| 2 | 14 | 2 | 68 59 |
| 2 | 15 | 2 | 78 69 |
| 2 | 16 | 1 | 79 |
| 2 | 17 | 1 | 89 |
| 3 | 6 | 1 | 123 |
| 3 | 7 | 1 | 124 |
| 3 | 8 | 2 | 134 125 |
| 3 | 9 | 3 | 234 135 126 |
| 3 | 10 | 4 | 235 145 136 127 |
| 3 | 11 | 5 | 245 236 146 137 128 |
| 3 | 12 | 7 | 345 246 156 237 147 138 129 |
| 3 | 13 | 7 | 346 256 247 157 238 148 139 |
| 3 | 14 | 8 | 356 347 257 167 248 158 239 149 |
| 3 | 15 | 8 | 456 357 267 348 258 168 249 159 |
| 3 | 16 | 8 | 457 367 358 268 178 349 259 169 |
| 3 | 17 | 7 | 467 458 368 278 359 269 179 |
| 3 | 18 | 7 | 567 468 378 459 369 279 189 |
| 3 | 19 | 5 | 568 478 469 379 289 |
| 3 | 20 | 4 | 578 569 479 389 |
| 3 | 21 | 3 | 678 579 489 |
| 3 | 22 | 2 | 679 589 |
| 3 | 23 | 1 | 689 |
| 3 | 24 | 1 | 789 |
| 4 | 10 | 1 | 1234 |
| 4 | 11 | 1 | 1235 |
| 4 | 12 | 2 | 1245 1236 |
| 4 | 13 | 3 | 1345 1246 1237 |
| 4 | 14 | 5 | 2345 1346 1256 1247 1238 |
| 4 | 15 | 6 | 2346 1356 1347 1257 1248 1239 |
| 4 | 16 | 8 | 2356 1456 2347 1357 1267 1348 1258 1249 |
| 4 | 17 | 9 | 2456 2357 1457 1367 2348 1358 1268 1349 1259 |
| 4 | 18 | 11 | 3456 2457 2367 1467 2358 1458 1368 1278 2349 1359 1269 |
| 4 | 19 | 11 | 3457 2467 1567 2458 2368 1468 1378 2359 1459 1369 1279 |
| 4 | 20 | 12 | 3467 2567 3458 2468 1568 2378 1478 2459 2369 1469 1379 1289 |
| 4 | 21 | 11 | 3567 3468 2568 2478 1578 3459 2469 1569 2379 1479 1389 |
| 4 | 22 | 11 | 4567 3568 3478 2578 1678 3469 2569 2479 1579 2389 1489 |
| 4 | 23 | 9 | 4568 3578 2678 3569 3479 2579 1679 2489 1589 |
| 4 | 24 | 8 | 4578 3678 4569 3579 2679 3489 2589 1689 |
| 4 | 25 | 6 | 4678 4579 3679 3589 2689 1789 |
| 4 | 26 | 5 | 5678 4679 4589 3689 2789 |
| 4 | 27 | 3 | 5679 4689 3789 |
| 4 | 28 | 2 | 5689 4789 |
| 4 | 29 | 1 | 5789 |
| 4 | 30 | 1 | 6789 |
| 5 | 15 | 1 | 12345 |
| 5 | 16 | 1 | 12346 |
| 5 | 17 | 2 | 12356 12347 |
| 5 | 18 | 3 | 12456 12357 12348 |
| 5 | 19 | 5 | 13456 12457 12367 12358 12349 |
| 5 | 20 | 6 | 23456 13457 12467 12458 12368 12359 |
| 5 | 21 | 8 | 23457 13467 12567 13458 12468 12378 12459 12369 |
| 5 | 22 | 9 | 23467 13567 23458 13468 12568 12478 13459 12469 12379 |
| 5 | 23 | 11 | 23567 14567 23468 13568 13478 12578 23459 13469 12569 12479 12389 |
| 5 | 24 | 11 | 24567 23568 14568 23478 13578 12678 23469 13569 13479 12579 12489 |
| 5 | 25 | 12 | 34567 24568 23578 14578 13678 23569 14569 23479 13579 12679 13489 12589 |
| 5 | 26 | 11 | 34568 24578 23678 14678 24569 23579 14579 13679 23489 13589 12689 |
| 5 | 27 | 11 | 34578 24678 15678 34569 24579 23679 14679 23589 14589 13689 12789 |
| 5 | 28 | 9 | 34678 25678 34579 24679 15679 24589 23689 14689 13789 |
| 5 | 29 | 8 | 35678 34679 25679 34589 24689 15689 23789 14789 |
| 5 | 30 | 6 | 45678 35679 34689 25689 24789 15789 |
| 5 | 31 | 5 | 45679 35689 34789 25789 16789 |
| 5 | 32 | 3 | 45689 35789 26789 |
| 5 | 33 | 2 | 45789 36789 |
| 5 | 34 | 1 | 46789 |
| 5 | 35 | 1 | 56789 |
How to use
- Press New puzzle, then tap a cell and type a digit from 1 to 9.
- Find the forced clues first — the highest and lowest sums for each run length.
- Digits cannot repeat within a run, which is what makes those clues forced.
- Use the combination table below when a clue has more than one option.
Frequently asked questions
What is Kakuro?
A crossword built from sums. Each run of white cells carries a clue giving the total of the digits in it, digits run from 1 to 9, and no digit may repeat within a single run. The clue for a run going across is written in the top right of the block before it, and the clue for a run going down in the bottom left.
How many clues can be made only one way?
Forty-three of the one hundred and twenty-nine possible clues, which is a third of them. A three-cell run summing to 6 is 1+2+3 and nothing else; a two-cell run summing to 17 is 8+9. Recognising those is where every solve starts, and it is why experienced players appear to fill in whole sections without any visible working.
Where are the forced clues?
At the extremes, and in a very regular pattern: at every run length from two to seven there are exactly four — the lowest sum and the next one up, the highest and the next one down. The smallest run of k digits is 1+2+...+k, and the only way to make one more is to push the top digit up by one. The same argument runs downward from the top, so the forced band is exactly two deep at each end.
What is the hardest clue to narrow down?
A five-cell run summing to 25, which can be made twelve different ways — the most of any clue. Four-cell runs summing to 20 are equally awkward. In general the sums nearest the middle of a run's range have the most combinations, which is the mirror image of why the extremes have only one.
Can a digit repeat in a Kakuro puzzle?
Not within one run, which is the rule that makes the whole thing work. It can and often does repeat elsewhere in the grid, including in a crossing run — two runs that share a cell each contain that digit once, but their other cells are unrelated. Reading the rule as "no repeats anywhere" makes many puzzles look unsolvable.
Does every puzzle here have exactly one answer?
Yes, and it is checked rather than intended. Every generated puzzle is passed to a solver that counts solutions and stops at two; anything with a second answer is thrown away and regenerated. That check matters more than it sounds: the first grid layouts drafted for this tool looked completely ordinary and admitted thousands of answers each.
Why are the grids small?
Because they are the ones that are provably unique. The layouts here were found by searching random block patterns for those where every run is two or three cells and at least one filling has a single solution. Open grids with long runs are barely constrained by their sums, so they look like proper Kakuro and are not.
Why does the grid have a border of blocks?
Because a clue is written in the block immediately before its run, so a run starting at the very edge would have nowhere to put one. Real Kakuro grids carry that border for the same reason. It looks like padding and is actually load-bearing.
What is a good approach to solving?
Find every forced clue first and pencil those digits in. Then look for crossings between a forced run and a nearly forced one, since a single known digit often collapses a clue with two or three options down to one. Working outward from certainty beats working through the grid in reading order, which is how most people get stuck.
Is Kakuro related to Sudoku?
Only distantly. Both use the digits 1 to 9 with a no-repeat rule, but Sudoku is about position and Kakuro is about arithmetic — a Kakuro clue tells you the total, which no Sudoku clue does. Kakuro is much older in spirit, descending from the cross sums puzzles that appeared in American magazines decades before Sudoku reached the West.
Why does Check take my run off the leaderboard?
Because on a puzzle with exactly one answer, Check tells you which of your entries are wrong — which turns guessing into a method. Fill the grid, press Check, correct whatever is flagged, repeat. That is solving by oracle rather than by arithmetic, so it is worth having as a learning aid and not worth ranking.
Does this store anything?
Only if you post a time. The puzzle is generated and solved entirely in your browser, nothing is uploaded, and nothing is kept between puzzles unless you choose to put a time on the global board, which sends the name you type and the number.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.