Kakuro Combinations Chart
In Kakuro, every run of white cells must add up to its clue using distinct digits from 1 to 9. That constraint means many clue-and-length pairs can be filled only one way — and recognising those forced combinations on sight is the fastest way to break a puzzle open.
This chart lists, for runs of 2 to 9 cells, every clue sum and the exact digit combinations that reach it. Sums with a single combination are the "magic blocks" worth memorising; the rest you narrow down with crossing runs. For a worked walkthrough, see our guide to Kakuro combinations in the Learn section.
Kakuro combinations answer one question: which digits can fill a run? In Kakuro, each clue gives the sum of a horizontal or vertical run of white cells, and every run uses distinct digits from 1 to 9 — a digit never repeats within a run. For any sum and run length, the set of valid digit combinations is fixed, and this page lists all of them.
The most valuable clues are the ones with a single combination. A run of 2 cells summing to 3 must be 1 and 2; a sum of 4 must be 1 and 3; sums of 16 and 17 must be 7+9 and 8+9. With 3 cells, a sum of 23 can only be 6+8+9, and a sum of 24 only 7+8+9. Where two such runs cross, the shared cell is often decided immediately.
Longer runs pin down digits too: a 9-cell run always sums to 45 and contains every digit once, an 8-cell run summing to 44 is missing exactly the digit 1, and in general an (L)-cell run at the minimum or maximum possible sum has only one combination. Cross-referencing a row clue against a column clue at their intersection is the core Kakuro technique.
How to find Kakuro combinations for any clue
- Select the clue sum shown next to the run.
- Select the run length — how many white cells the clue covers.
- Read every valid combination of distinct digits 1–9.
- Intersect the row combinations with the column combinations at each shared cell.
- Fill any cell where only one digit survives the intersection.
Runs of 2 cells
| Clue sum | Possible combinations |
|---|---|
| 3 | 1+2 |
| 4 | 1+3 |
| 5 | 1+4, 2+3 |
| 6 | 1+5, 2+4 |
| 7 | 1+6, 2+5, 3+4 |
| 8 | 1+7, 2+6, 3+5 |
| 9 | 1+8, 2+7, 3+6, 4+5 |
| 10 | 1+9, 2+8, 3+7, 4+6 |
| 11 | 2+9, 3+8, 4+7, 5+6 |
| 12 | 3+9, 4+8, 5+7 |
| 13 | 4+9, 5+8, 6+7 |
| 14 | 5+9, 6+8 |
| 15 | 6+9, 7+8 |
| 16 | 7+9 |
| 17 | 8+9 |
Runs of 3 cells
| Clue sum | Possible combinations |
|---|---|
| 6 | 1+2+3 |
| 7 | 1+2+4 |
| 8 | 1+2+5, 1+3+4 |
| 9 | 1+2+6, 1+3+5, 2+3+4 |
| 10 | 1+2+7, 1+3+6, 1+4+5, 2+3+5 |
| 11 | 1+2+8, 1+3+7, 1+4+6, 2+3+6, 2+4+5 |
| 12 | 1+2+9, 1+3+8, 1+4+7, 1+5+6, 2+3+7, 2+4+6, 3+4+5 |
| 13 | 1+3+9, 1+4+8, 1+5+7, 2+3+8, 2+4+7, 2+5+6, 3+4+6 |
| 14 | 1+4+9, 1+5+8, 1+6+7, 2+3+9, 2+4+8, 2+5+7, 3+4+7, 3+5+6 |
| 15 | 1+5+9, 1+6+8, 2+4+9, 2+5+8, 2+6+7, 3+4+8, 3+5+7, 4+5+6 |
| 16 | 1+6+9, 1+7+8, 2+5+9, 2+6+8, 3+4+9, 3+5+8, 3+6+7, 4+5+7 |
| 17 | 1+7+9, 2+6+9, 2+7+8, 3+5+9, 3+6+8, 4+5+8, 4+6+7 |
| 18 | 1+8+9, 2+7+9, 3+6+9, 3+7+8, 4+5+9, 4+6+8, 5+6+7 |
| 19 | 2+8+9, 3+7+9, 4+6+9, 4+7+8, 5+6+8 |
| 20 | 3+8+9, 4+7+9, 5+6+9, 5+7+8 |
| 21 | 4+8+9, 5+7+9, 6+7+8 |
| 22 | 5+8+9, 6+7+9 |
| 23 | 6+8+9 |
| 24 | 7+8+9 |
Runs of 4 cells
| Clue sum | Possible combinations |
|---|---|
| 10 | 1+2+3+4 |
| 11 | 1+2+3+5 |
| 12 | 1+2+3+6, 1+2+4+5 |
| 13 | 1+2+3+7, 1+2+4+6, 1+3+4+5 |
| 14 | 1+2+3+8, 1+2+4+7, 1+2+5+6, 1+3+4+6, 2+3+4+5 |
| 15 | 1+2+3+9, 1+2+4+8, 1+2+5+7, 1+3+4+7, 1+3+5+6, 2+3+4+6 |
| 16 | 1+2+4+9, 1+2+5+8, 1+2+6+7, 1+3+4+8, 1+3+5+7, 1+4+5+6, 2+3+4+7, 2+3+5+6 |
| 17 | 1+2+5+9, 1+2+6+8, 1+3+4+9, 1+3+5+8, 1+3+6+7, 1+4+5+7, 2+3+4+8, 2+3+5+7, 2+4+5+6 |
| 18 | 1+2+6+9, 1+2+7+8, 1+3+5+9, 1+3+6+8, 1+4+5+8, 1+4+6+7, 2+3+4+9, 2+3+5+8, 2+3+6+7, 2+4+5+7, 3+4+5+6 |
| 19 | 1+2+7+9, 1+3+6+9, 1+3+7+8, 1+4+5+9, 1+4+6+8, 1+5+6+7, 2+3+5+9, 2+3+6+8, 2+4+5+8, 2+4+6+7, 3+4+5+7 |
| 20 | 1+2+8+9, 1+3+7+9, 1+4+6+9, 1+4+7+8, 1+5+6+8, 2+3+6+9, 2+3+7+8, 2+4+5+9, 2+4+6+8, 2+5+6+7, 3+4+5+8, 3+4+6+7 |
| 21 | 1+3+8+9, 1+4+7+9, 1+5+6+9, 1+5+7+8, 2+3+7+9, 2+4+6+9, 2+4+7+8, 2+5+6+8, 3+4+5+9, 3+4+6+8, 3+5+6+7 |
| 22 | 1+4+8+9, 1+5+7+9, 1+6+7+8, 2+3+8+9, 2+4+7+9, 2+5+6+9, 2+5+7+8, 3+4+6+9, 3+4+7+8, 3+5+6+8, 4+5+6+7 |
| 23 | 1+5+8+9, 1+6+7+9, 2+4+8+9, 2+5+7+9, 2+6+7+8, 3+4+7+9, 3+5+6+9, 3+5+7+8, 4+5+6+8 |
| 24 | 1+6+8+9, 2+5+8+9, 2+6+7+9, 3+4+8+9, 3+5+7+9, 3+6+7+8, 4+5+6+9, 4+5+7+8 |
| 25 | 1+7+8+9, 2+6+8+9, 3+5+8+9, 3+6+7+9, 4+5+7+9, 4+6+7+8 |
| 26 | 2+7+8+9, 3+6+8+9, 4+5+8+9, 4+6+7+9, 5+6+7+8 |
| 27 | 3+7+8+9, 4+6+8+9, 5+6+7+9 |
| 28 | 4+7+8+9, 5+6+8+9 |
| 29 | 5+7+8+9 |
| 30 | 6+7+8+9 |
Runs of 5 cells
| Clue sum | Possible combinations |
|---|---|
| 15 | 1+2+3+4+5 |
| 16 | 1+2+3+4+6 |
| 17 | 1+2+3+4+7, 1+2+3+5+6 |
| 18 | 1+2+3+4+8, 1+2+3+5+7, 1+2+4+5+6 |
| 19 | 1+2+3+4+9, 1+2+3+5+8, 1+2+3+6+7, 1+2+4+5+7, 1+3+4+5+6 |
| 20 | 1+2+3+5+9, 1+2+3+6+8, 1+2+4+5+8, 1+2+4+6+7, 1+3+4+5+7, 2+3+4+5+6 |
| 21 | 1+2+3+6+9, 1+2+3+7+8, 1+2+4+5+9, 1+2+4+6+8, 1+2+5+6+7, 1+3+4+5+8, 1+3+4+6+7, 2+3+4+5+7 |
| 22 | 1+2+3+7+9, 1+2+4+6+9, 1+2+4+7+8, 1+2+5+6+8, 1+3+4+5+9, 1+3+4+6+8, 1+3+5+6+7, 2+3+4+5+8, 2+3+4+6+7 |
| 23 | 1+2+3+8+9, 1+2+4+7+9, 1+2+5+6+9, 1+2+5+7+8, 1+3+4+6+9, 1+3+4+7+8, 1+3+5+6+8, 1+4+5+6+7, 2+3+4+5+9, 2+3+4+6+8, 2+3+5+6+7 |
| 24 | 1+2+4+8+9, 1+2+5+7+9, 1+2+6+7+8, 1+3+4+7+9, 1+3+5+6+9, 1+3+5+7+8, 1+4+5+6+8, 2+3+4+6+9, 2+3+4+7+8, 2+3+5+6+8, 2+4+5+6+7 |
| 25 | 1+2+5+8+9, 1+2+6+7+9, 1+3+4+8+9, 1+3+5+7+9, 1+3+6+7+8, 1+4+5+6+9, 1+4+5+7+8, 2+3+4+7+9, 2+3+5+6+9, 2+3+5+7+8, 2+4+5+6+8, 3+4+5+6+7 |
| 26 | 1+2+6+8+9, 1+3+5+8+9, 1+3+6+7+9, 1+4+5+7+9, 1+4+6+7+8, 2+3+4+8+9, 2+3+5+7+9, 2+3+6+7+8, 2+4+5+6+9, 2+4+5+7+8, 3+4+5+6+8 |
| 27 | 1+2+7+8+9, 1+3+6+8+9, 1+4+5+8+9, 1+4+6+7+9, 1+5+6+7+8, 2+3+5+8+9, 2+3+6+7+9, 2+4+5+7+9, 2+4+6+7+8, 3+4+5+6+9, 3+4+5+7+8 |
| 28 | 1+3+7+8+9, 1+4+6+8+9, 1+5+6+7+9, 2+3+6+8+9, 2+4+5+8+9, 2+4+6+7+9, 2+5+6+7+8, 3+4+5+7+9, 3+4+6+7+8 |
| 29 | 1+4+7+8+9, 1+5+6+8+9, 2+3+7+8+9, 2+4+6+8+9, 2+5+6+7+9, 3+4+5+8+9, 3+4+6+7+9, 3+5+6+7+8 |
| 30 | 1+5+7+8+9, 2+4+7+8+9, 2+5+6+8+9, 3+4+6+8+9, 3+5+6+7+9, 4+5+6+7+8 |
| 31 | 1+6+7+8+9, 2+5+7+8+9, 3+4+7+8+9, 3+5+6+8+9, 4+5+6+7+9 |
| 32 | 2+6+7+8+9, 3+5+7+8+9, 4+5+6+8+9 |
| 33 | 3+6+7+8+9, 4+5+7+8+9 |
| 34 | 4+6+7+8+9 |
| 35 | 5+6+7+8+9 |
Runs of 6 cells
| Clue sum | Possible combinations |
|---|---|
| 21 | 1+2+3+4+5+6 |
| 22 | 1+2+3+4+5+7 |
| 23 | 1+2+3+4+5+8, 1+2+3+4+6+7 |
| 24 | 1+2+3+4+5+9, 1+2+3+4+6+8, 1+2+3+5+6+7 |
| 25 | 1+2+3+4+6+9, 1+2+3+4+7+8, 1+2+3+5+6+8, 1+2+4+5+6+7 |
| 26 | 1+2+3+4+7+9, 1+2+3+5+6+9, 1+2+3+5+7+8, 1+2+4+5+6+8, 1+3+4+5+6+7 |
| 27 | 1+2+3+4+8+9, 1+2+3+5+7+9, 1+2+3+6+7+8, 1+2+4+5+6+9, 1+2+4+5+7+8, 1+3+4+5+6+8, 2+3+4+5+6+7 |
| 28 | 1+2+3+5+8+9, 1+2+3+6+7+9, 1+2+4+5+7+9, 1+2+4+6+7+8, 1+3+4+5+6+9, 1+3+4+5+7+8, 2+3+4+5+6+8 |
| 29 | 1+2+3+6+8+9, 1+2+4+5+8+9, 1+2+4+6+7+9, 1+2+5+6+7+8, 1+3+4+5+7+9, 1+3+4+6+7+8, 2+3+4+5+6+9, 2+3+4+5+7+8 |
| 30 | 1+2+3+7+8+9, 1+2+4+6+8+9, 1+2+5+6+7+9, 1+3+4+5+8+9, 1+3+4+6+7+9, 1+3+5+6+7+8, 2+3+4+5+7+9, 2+3+4+6+7+8 |
| 31 | 1+2+4+7+8+9, 1+2+5+6+8+9, 1+3+4+6+8+9, 1+3+5+6+7+9, 1+4+5+6+7+8, 2+3+4+5+8+9, 2+3+4+6+7+9, 2+3+5+6+7+8 |
| 32 | 1+2+5+7+8+9, 1+3+4+7+8+9, 1+3+5+6+8+9, 1+4+5+6+7+9, 2+3+4+6+8+9, 2+3+5+6+7+9, 2+4+5+6+7+8 |
| 33 | 1+2+6+7+8+9, 1+3+5+7+8+9, 1+4+5+6+8+9, 2+3+4+7+8+9, 2+3+5+6+8+9, 2+4+5+6+7+9, 3+4+5+6+7+8 |
| 34 | 1+3+6+7+8+9, 1+4+5+7+8+9, 2+3+5+7+8+9, 2+4+5+6+8+9, 3+4+5+6+7+9 |
| 35 | 1+4+6+7+8+9, 2+3+6+7+8+9, 2+4+5+7+8+9, 3+4+5+6+8+9 |
| 36 | 1+5+6+7+8+9, 2+4+6+7+8+9, 3+4+5+7+8+9 |
| 37 | 2+5+6+7+8+9, 3+4+6+7+8+9 |
| 38 | 3+5+6+7+8+9 |
| 39 | 4+5+6+7+8+9 |
Runs of 7 cells
| Clue sum | Possible combinations |
|---|---|
| 28 | 1+2+3+4+5+6+7 |
| 29 | 1+2+3+4+5+6+8 |
| 30 | 1+2+3+4+5+6+9, 1+2+3+4+5+7+8 |
| 31 | 1+2+3+4+5+7+9, 1+2+3+4+6+7+8 |
| 32 | 1+2+3+4+5+8+9, 1+2+3+4+6+7+9, 1+2+3+5+6+7+8 |
| 33 | 1+2+3+4+6+8+9, 1+2+3+5+6+7+9, 1+2+4+5+6+7+8 |
| 34 | 1+2+3+4+7+8+9, 1+2+3+5+6+8+9, 1+2+4+5+6+7+9, 1+3+4+5+6+7+8 |
| 35 | 1+2+3+5+7+8+9, 1+2+4+5+6+8+9, 1+3+4+5+6+7+9, 2+3+4+5+6+7+8 |
| 36 | 1+2+3+6+7+8+9, 1+2+4+5+7+8+9, 1+3+4+5+6+8+9, 2+3+4+5+6+7+9 |
| 37 | 1+2+4+6+7+8+9, 1+3+4+5+7+8+9, 2+3+4+5+6+8+9 |
| 38 | 1+2+5+6+7+8+9, 1+3+4+6+7+8+9, 2+3+4+5+7+8+9 |
| 39 | 1+3+5+6+7+8+9, 2+3+4+6+7+8+9 |
| 40 | 1+4+5+6+7+8+9, 2+3+5+6+7+8+9 |
| 41 | 2+4+5+6+7+8+9 |
| 42 | 3+4+5+6+7+8+9 |
Runs of 8 cells
| Clue sum | Possible combinations |
|---|---|
| 36 | 1+2+3+4+5+6+7+8 |
| 37 | 1+2+3+4+5+6+7+9 |
| 38 | 1+2+3+4+5+6+8+9 |
| 39 | 1+2+3+4+5+7+8+9 |
| 40 | 1+2+3+4+6+7+8+9 |
| 41 | 1+2+3+5+6+7+8+9 |
| 42 | 1+2+4+5+6+7+8+9 |
| 43 | 1+3+4+5+6+7+8+9 |
| 44 | 2+3+4+5+6+7+8+9 |
Runs of 9 cells
| Clue sum | Possible combinations |
|---|---|
| 45 | 1+2+3+4+5+6+7+8+9 |
Frequently asked questions
- Can digits repeat in a Kakuro run?
- No. Every run of white cells must use distinct digits from 1 to 9. The same digit can appear elsewhere in the grid, but never twice within one run.
- Which Kakuro sums have only one combination?
- For 2 cells: 3 (1+2), 4 (1+3), 16 (7+9), 17 (8+9). For 3 cells: 6 (1+2+3), 7 (1+2+4), 23 (6+8+9), 24 (7+8+9). Every run length has such unique sums at its minimum and maximum, and they are the best places to start solving.
- What are the combinations for 23 in Kakuro?
- Over 3 cells, 23 has exactly one combination: 6+8+9. Over 4 cells there are several, such as 1+5+8+9 and 2+5+7+9; over 5 cells more still. Length matters as much as the sum.
- What is the smallest and largest sum for a Kakuro run?
- A run of L cells sums to at least 1+2+…+L and at most 9+8+…+(10−L). For example, a 4-cell run ranges from 10 (1+2+3+4) to 30 (6+7+8+9), and a 9-cell run always sums to exactly 45.
- Is using a Kakuro combinations lookup cheating?
- Most solvers treat it like a times table: the combinations are fixed mathematical facts, and looking them up lets you focus on the real puzzle — the logic of intersecting runs.
Kakuro · Learn · Kakuro Combinations: The Number Blocks That Crack the Grid