Kakuro Combination Reference
Every valid set of distinct digits (1–9) for each run length and sum value. Rows highlighted in blue mark forced (unique) combinations — when only one combination exists, those digits must appear in the run regardless of cell order.
Use this table alongside the Sum Singles and Combination Pruning guides. If you want to filter a single clue interactively, open the Kakuro helper. Jump to a run length: 2· 3· 4· 5· 6· 7· 8· 9.
2-Cell Runs
Sum range: 3–17 · 4 forced combinations
| Sum | Valid combinations | Count |
|---|---|---|
| 3★ | {1+2} | 1 |
| 4★ | {1+3} | 1 |
| 5 | {1+4}{2+3} | 2 |
| 6 | {1+5}{2+4} | 2 |
| 7 | {1+6}{2+5}{3+4} | 3 |
| 8 | {1+7}{2+6}{3+5} | 3 |
| 9 | {1+8}{2+7}{3+6}{4+5} | 4 |
| 10 | {1+9}{2+8}{3+7}{4+6} | 4 |
| 11 | {2+9}{3+8}{4+7}{5+6} | 4 |
| 12 | {3+9}{4+8}{5+7} | 3 |
| 13 | {4+9}{5+8}{6+7} | 3 |
| 14 | {5+9}{6+8} | 2 |
| 15 | {6+9}{7+8} | 2 |
| 16★ | {7+9} | 1 |
| 17★ | {8+9} | 1 |
3-Cell Runs
Sum range: 6–24 · 4 forced combinations
| Sum | Valid combinations | Count |
|---|---|---|
| 6★ | {1+2+3} | 1 |
| 7★ | {1+2+4} | 1 |
| 8 | {1+2+5}{1+3+4} | 2 |
| 9 | {1+2+6}{1+3+5}{2+3+4} | 3 |
| 10 | {1+2+7}{1+3+6}{1+4+5}{2+3+5} | 4 |
| 11 | {1+2+8}{1+3+7}{1+4+6}{2+3+6}{2+4+5} | 5 |
| 12 | {1+2+9}{1+3+8}{1+4+7}{1+5+6}{2+3+7}{2+4+6}{3+4+5} | 7 |
| 13 | {1+3+9}{1+4+8}{1+5+7}{2+3+8}{2+4+7}{2+5+6}{3+4+6} | 7 |
| 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} | 8 |
| 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} | 8 |
| 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} | 8 |
| 17 | {1+7+9}{2+6+9}{2+7+8}{3+5+9}{3+6+8}{4+5+8}{4+6+7} | 7 |
| 18 | {1+8+9}{2+7+9}{3+6+9}{3+7+8}{4+5+9}{4+6+8}{5+6+7} | 7 |
| 19 | {2+8+9}{3+7+9}{4+6+9}{4+7+8}{5+6+8} | 5 |
| 20 | {3+8+9}{4+7+9}{5+6+9}{5+7+8} | 4 |
| 21 | {4+8+9}{5+7+9}{6+7+8} | 3 |
| 22 | {5+8+9}{6+7+9} | 2 |
| 23★ | {6+8+9} | 1 |
| 24★ | {7+8+9} | 1 |
4-Cell Runs
Sum range: 10–30 · 4 forced combinations
| Sum | Valid combinations | Count |
|---|---|---|
| 10★ | {1+2+3+4} | 1 |
| 11★ | {1+2+3+5} | 1 |
| 12 | {1+2+3+6}{1+2+4+5} | 2 |
| 13 | {1+2+3+7}{1+2+4+6}{1+3+4+5} | 3 |
| 14 | {1+2+3+8}{1+2+4+7}{1+2+5+6}{1+3+4+6}{2+3+4+5} | 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} | 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} | 8 |
| 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} | 9 |
| 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} | 11 |
| 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} | 11 |
| 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} | 12 |
| 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} | 11 |
| 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} | 11 |
| 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} | 9 |
| 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} | 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} | 6 |
| 26 | {2+7+8+9}{3+6+8+9}{4+5+8+9}{4+6+7+9}{5+6+7+8} | 5 |
| 27 | {3+7+8+9}{4+6+8+9}{5+6+7+9} | 3 |
| 28 | {4+7+8+9}{5+6+8+9} | 2 |
| 29★ | {5+7+8+9} | 1 |
| 30★ | {6+7+8+9} | 1 |
5-Cell Runs
Sum range: 15–35 · 4 forced combinations
| Sum | Valid combinations | Count |
|---|---|---|
| 15★ | {1+2+3+4+5} | 1 |
| 16★ | {1+2+3+4+6} | 1 |
| 17 | {1+2+3+4+7}{1+2+3+5+6} | 2 |
| 18 | {1+2+3+4+8}{1+2+3+5+7}{1+2+4+5+6} | 3 |
| 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} | 5 |
| 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} | 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} | 8 |
| 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} | 9 |
| 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} | 11 |
| 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} | 11 |
| 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} | 12 |
| 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} | 11 |
| 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} | 11 |
| 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} | 9 |
| 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} | 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} | 6 |
| 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} | 5 |
| 32 | {2+6+7+8+9}{3+5+7+8+9}{4+5+6+8+9} | 3 |
| 33 | {3+6+7+8+9}{4+5+7+8+9} | 2 |
| 34★ | {4+6+7+8+9} | 1 |
| 35★ | {5+6+7+8+9} | 1 |
6-Cell Runs
Sum range: 21–39 · 4 forced combinations
| Sum | Valid combinations | Count |
|---|---|---|
| 21★ | {1+2+3+4+5+6} | 1 |
| 22★ | {1+2+3+4+5+7} | 1 |
| 23 | {1+2+3+4+5+8}{1+2+3+4+6+7} | 2 |
| 24 | {1+2+3+4+5+9}{1+2+3+4+6+8}{1+2+3+5+6+7} | 3 |
| 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} | 4 |
| 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} | 5 |
| 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} | 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} | 7 |
| 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} | 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} | 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} | 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} | 7 |
| 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} | 7 |
| 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} | 5 |
| 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} | 4 |
| 36 | {1+5+6+7+8+9}{2+4+6+7+8+9}{3+4+5+7+8+9} | 3 |
| 37 | {2+5+6+7+8+9}{3+4+6+7+8+9} | 2 |
| 38★ | {3+5+6+7+8+9} | 1 |
| 39★ | {4+5+6+7+8+9} | 1 |
7-Cell Runs
Sum range: 28–42 · 4 forced combinations
| Sum | Valid combinations | Count |
|---|---|---|
| 28★ | {1+2+3+4+5+6+7} | 1 |
| 29★ | {1+2+3+4+5+6+8} | 1 |
| 30 | {1+2+3+4+5+6+9}{1+2+3+4+5+7+8} | 2 |
| 31 | {1+2+3+4+5+7+9}{1+2+3+4+6+7+8} | 2 |
| 32 | {1+2+3+4+5+8+9}{1+2+3+4+6+7+9}{1+2+3+5+6+7+8} | 3 |
| 33 | {1+2+3+4+6+8+9}{1+2+3+5+6+7+9}{1+2+4+5+6+7+8} | 3 |
| 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} | 4 |
| 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} | 4 |
| 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} | 4 |
| 37 | {1+2+4+6+7+8+9}{1+3+4+5+7+8+9}{2+3+4+5+6+8+9} | 3 |
| 38 | {1+2+5+6+7+8+9}{1+3+4+6+7+8+9}{2+3+4+5+7+8+9} | 3 |
| 39 | {1+3+5+6+7+8+9}{2+3+4+6+7+8+9} | 2 |
| 40 | {1+4+5+6+7+8+9}{2+3+5+6+7+8+9} | 2 |
| 41★ | {2+4+5+6+7+8+9} | 1 |
| 42★ | {3+4+5+6+7+8+9} | 1 |
8-Cell Runs
Sum range: 36–44 · 9 forced combinations
| Sum | Valid combinations | Count |
|---|---|---|
| 36★ | {1+2+3+4+5+6+7+8} | 1 |
| 37★ | {1+2+3+4+5+6+7+9} | 1 |
| 38★ | {1+2+3+4+5+6+8+9} | 1 |
| 39★ | {1+2+3+4+5+7+8+9} | 1 |
| 40★ | {1+2+3+4+6+7+8+9} | 1 |
| 41★ | {1+2+3+5+6+7+8+9} | 1 |
| 42★ | {1+2+4+5+6+7+8+9} | 1 |
| 43★ | {1+3+4+5+6+7+8+9} | 1 |
| 44★ | {2+3+4+5+6+7+8+9} | 1 |
9-Cell Runs
Sum range: 45–45 · 1 forced combination
| Sum | Valid combinations | Count |
|---|---|---|
| 45★ | {1+2+3+4+5+6+7+8+9} | 1 |
Summary: forced combinations by run length
A forced combination has exactly one valid digit set for the given sum. Memorise these first — they require no deduction and immediately fill cells.
| Run length | Sum range | Forced sums | Forced / Total |
|---|---|---|---|
| 2 cells | 3–17 | 3, 4, 16, 17 | 4 / 15 |
| 3 cells | 6–24 | 6, 7, 23, 24 | 4 / 19 |
| 4 cells | 10–30 | 10, 11, 29, 30 | 4 / 21 |
| 5 cells | 15–35 | 15, 16, 34, 35 | 4 / 21 |
| 6 cells | 21–39 | 21, 22, 38, 39 | 4 / 19 |
| 7 cells | 28–42 | 28, 29, 41, 42 | 4 / 15 |
| 8 cells | 36–44 | 36, 37, 38, 39, 40, 41, 42, 43, 44 | 9 / 9 |
| 9 cells | 45–45 | 45 | 1 / 1 |
How to use this table
Step 1 — Find your run length and sum
Count the white cells in the run (that is the run length) and read the clue value (that is the sum). Jump to the matching section above.
Step 2 — Read the valid combinations
Each set in curly braces is one possible group of digits. For example, a 3-cell run summing to 9 can be solved with {1+2+6}, {1+3+5}, or {2+3+4}.
Step 3 — Eliminate via crossing runs
If a crossing run already contains the digit 6, eliminate any combination that includes 6. Repeat across all intersections until one combination remains.
Forced combinations (★)
When only one combination is listed, the digit set is forced. You know which digits appear — you just need to determine their order from the crossing runs. Start your solve with these.
Quick-reference: the most useful forced combinations
These appear constantly in easy and medium puzzles. Memorise these eight and your first scan of any grid will unlock cells immediately.
| Run | Sum | Forced digits |
|---|---|---|
| 2 cells | 3 | {1, 2} |
| 2 cells | 17 | {8, 9} |
| 3 cells | 6 | {1, 2, 3} |
| 3 cells | 24 | {7, 8, 9} |
| 4 cells | 10 | {1, 2, 3, 4} |
| 4 cells | 30 | {6, 7, 8, 9} |
| 5 cells | 15 | {1, 2, 3, 4, 5} |
| 5 cells | 35 | {5, 6, 7, 8, 9} |
Kakuro combinations FAQ
- What are Kakuro combinations?
- Kakuro combinations are the sets of distinct digits (1–9) that can fill a run of cells and sum to the clue value. For example, a 3-cell run summing to 9 can be filled with {1,2,6}, {1,3,5}, or {2,3,4}. No digit may repeat within a run.
- What is a forced Kakuro combination?
- A forced combination exists when only one set of digits can satisfy a given run length and sum. For example, a 2-cell run summing to 3 must contain {1,2}. Solving forced combinations first is the fastest way to make progress on any Kakuro puzzle.
- How many combinations are there for a 3-cell run summing to 16?
- Five: {1,6,9}, {2,5,9}, {3,4,9}, {3,6,7}, and {4,5,7}. Use constraints from crossing runs to eliminate combinations containing digits already placed.
- What is the minimum and maximum sum for a Kakuro run?
- For a run of n cells, the minimum sum is n(n+1)/2 and the maximum is n(19−n)/2. A 4-cell run ranges from 10 (1+2+3+4) to 30 (6+7+8+9). Any clue outside this range for its run length is impossible.
- How do I use a Kakuro combination table to solve a puzzle?
- Find your run length and clue sum in the table, list all valid combinations, then eliminate any that include a digit already placed in a crossing run. If only one combination remains, those are your digits. If every remaining combination shares a digit in one cell position, that digit is placed there regardless of other choices. The Kakuro helper automates this filtering interactively.