Kakuro Combinations
In Kakuro, every run must be filled with distinct digits (1–9) that add up to the clue value. The set of digits you use is called a combination. Understanding which combinations are possible — and which are forced — is the core skill that separates fast solvers from slow ones.
This page explains how Kakuro number combinations work, shows the most important combinations chart, and links to the full reference table and interactive helper.
How Kakuro number combinations work
Each run in a Kakuro grid has two properties: its length (the number of white cells) and its clue (the target sum written in the grey header cell). To fill the run correctly you must find a set of that many distinct digits from 1 to 9 that add up to the clue.
Because digits cannot repeat within a run, the number of valid combinations is always limited. A 2-cell run can have at most 4 valid combinations for any given sum; a 3-cell run at most 6; and as runs get longer, the space of possibilities narrows again near the minimum and maximum sums.
The key insight is that fewer valid combinations means more constraint. When a run has only one valid combination — called a forced or unique combination — you know exactly which digits go there. When it has two or three, crossing runs let you eliminate the impossible ones. Working from the most constrained runs outward is the standard Kakuro strategy.
Kakuro combinations chart — forced combinations
These are the forced (unique) Kakuro combinations for runs of 2–5 cells. A forced combination has exactly one valid digit set, so you can place all those digits immediately — no crossing-run logic needed. Memorise these first.
| Cells | Sum | Forced digits | Note |
|---|---|---|---|
| 2 cells | 3 | {1, 2} | Minimum 2-cell sum |
| 4 | {1, 3} | ||
| 16 | {7, 9} | ||
| 17 | {8, 9} | Maximum 2-cell sum | |
| 3 cells | 6 | {1, 2, 3} | Minimum 3-cell sum |
| 7 | {1, 2, 4} | ||
| 23 | {6, 8, 9} | ||
| 24 | {7, 8, 9} | Maximum 3-cell sum | |
| 4 cells | 10 | {1, 2, 3, 4} | Minimum 4-cell sum |
| 11 | {1, 2, 3, 5} | ||
| 29 | {5, 7, 8, 9} | ||
| 30 | {6, 7, 8, 9} | Maximum 4-cell sum | |
| 5 cells | 15 | {1, 2, 3, 4, 5} | Minimum 5-cell sum |
| 16 | {1, 2, 3, 4, 6} | ||
| 34 | {4, 6, 7, 8, 9} | ||
| 35 | {5, 6, 7, 8, 9} | Maximum 5-cell sum |
For the complete combinations table covering all sums and all run lengths (2–9 cells), see the full Kakuro combination reference. For interactive lookup — type in your clue and get the valid sets instantly — use the Kakuro helper.
Kakuro combinations table — counts by run length
The table below shows the valid sum range and number of distinct sums for each run length. Longer runs have a narrower percentage of forced sums — most of the constraint comes from crossing runs, not the clue alone.
| Run length | Min sum | Max sum | Distinct sums |
|---|---|---|---|
| 2 cells | 3 | 17 | 15 |
| 3 cells | 6 | 24 | 19 |
| 4 cells | 10 | 30 | 21 |
| 5 cells | 15 | 35 | 21 |
| 6 cells | 21 | 39 | 19 |
| 7 cells | 28 | 42 | 15 |
| 8 cells | 36 | 44 | 9 |
| 9 cells | 45 | 45 | 1 |
Note: a 9-cell run has only one possible sum (45 = 1+2+3+4+5+6+7+8+9), so it is always forced.
2-cell Kakuro combinations chart
Two-cell runs are the most constrained — sums range from 3 to 17. Sums 3, 4, 16, and 17 are forced (★); all others have two or more combinations.
| Sum | Valid 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} |
3-cell Kakuro combinations chart
Three-cell runs span sums 6–24. Sums 6, 7, 23, and 24 are forced (★); the middle sums have many combinations.
| Sum | Valid 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} |
For 4-cell through 9-cell combination tables, see the full Kakuro combinations reference.
Using Kakuro combinations to solve puzzles
- 1 Identify forced combinations first. Scan every clue and check whether it has only one valid digit set (see the chart above). Place those digits immediately — they give you free cells with zero deduction.
- 2 List candidates for constrained runs. For clues with two or three valid combinations, write each set down. Any digit that appears in every combination for that run is guaranteed — use it to eliminate candidates in the crossing runs.
- 3 Prune via intersections. Each white cell lies at the crossing of an across and a down run. If the across combination can never include digit 7, cross 7 off that cell's candidates for the down run too. This cascade shrinks the possibilities fast.
- 4 Use the combination reference for unfamiliar clues. When you hit an unusual sum/length pair, look it up in the full combinations table rather than re-computing from scratch.
For a deeper treatment of each technique, see the technique library, starting with Sum Singles and Combination Pruning.
Frequently asked questions
- What are Kakuro combinations?
- Kakuro combinations are the sets of distinct digits (1–9) that satisfy a clue — they add up to the target sum across the run's cells, with no digit repeated. For example, the combinations for a 2-cell clue of 7 are {1,6}, {2,5}, and {3,4}.
- What is a forced Kakuro combination?
- A forced combination is when only one digit set satisfies the clue. A 3-cell run summing to 6 can only be {1,2,3}. Forced combinations let you place digits immediately, with no need to analyse crossing runs.
- How many Kakuro combinations are there in total?
- Across all run lengths (2–9 cells) and all valid sum values, there are several hundred unique digit sets. The complete reference table lists all of them — you do not need to memorise them all. Focus on the forced combinations first; use the table for the rest.
- Which Kakuro combinations should I memorise?
- Memorise the forced combinations listed in the chart above — particularly the 2-cell sums (3, 4, 16, 17) and 3-cell sums (6, 7, 23, 24). These appear in almost every puzzle and give instant placements that unlock the rest of the grid.
- Where can I look up Kakuro number combinations quickly?
- Use the interactive Kakuro helper — enter your run length and clue value and see all valid combinations instantly. For a printable overview, the Kakuro cheat sheet covers the most common cases on one page.