Why some totals are special
In kakuro a run may not repeat a digit, so a run of k cells always uses k different digits from 1 to 9. For most totals there are several ways to do that. Two cells adding up to 10 can be 1+9, 2+8, 3+7 or 4+6. But the smallest and largest possible totals for each length have only one way, and so do a few just next to them.
Two cells can add up to anything from 3 (1+2) to 17 (8+9). Three cells range from 6 (1+2+3) to 24 (7+8+9). The ends of each range are forced, and the values just inside them usually are too.
The complete table
Below is every total that has exactly one combination, for every run length. The table was computed from all 511 possible digit sets, so nothing is missing.
| Cells | Total | Only digits |
|---|---|---|
| 2 | 3 | 1 + 2 |
| 2 | 4 | 1 + 3 |
| 2 | 16 | 7 + 9 |
| 2 | 17 | 8 + 9 |
| 3 | 6 | 1 + 2 + 3 |
| 3 | 7 | 1 + 2 + 4 |
| 3 | 23 | 6 + 8 + 9 |
| 3 | 24 | 7 + 8 + 9 |
| 4 | 10 | 1 + 2 + 3 + 4 |
| 4 | 11 | 1 + 2 + 3 + 5 |
| 4 | 29 | 5 + 7 + 8 + 9 |
| 4 | 30 | 6 + 7 + 8 + 9 |
| 5 | 15 | 1 + 2 + 3 + 4 + 5 |
| 5 | 16 | 1 + 2 + 3 + 4 + 6 |
| 5 | 34 | 4 + 6 + 7 + 8 + 9 |
| 5 | 35 | 5 + 6 + 7 + 8 + 9 |
| 6 | 21 | 1 + 2 + 3 + 4 + 5 + 6 |
| 6 | 22 | 1 + 2 + 3 + 4 + 5 + 7 |
| 6 | 38 | 3 + 5 + 6 + 7 + 8 + 9 |
| 6 | 39 | 4 + 5 + 6 + 7 + 8 + 9 |
| 7 | 28 | 1 + 2 + 3 + 4 + 5 + 6 + 7 |
| 7 | 29 | 1 + 2 + 3 + 4 + 5 + 6 + 8 |
| 7 | 41 | 2 + 4 + 5 + 6 + 7 + 8 + 9 |
| 7 | 42 | 3 + 4 + 5 + 6 + 7 + 8 + 9 |
| 8 | 36 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 |
| 8 | 37 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 9 |
| 8 | 38 | 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 |
| 8 | 39 | 1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 |
| 8 | 40 | 1 + 2 + 3 + 4 + 6 + 7 + 8 + 9 |
| 8 | 41 | 1 + 2 + 3 + 5 + 6 + 7 + 8 + 9 |
| 8 | 42 | 1 + 2 + 4 + 5 + 6 + 7 + 8 + 9 |
| 8 | 43 | 1 + 3 + 4 + 5 + 6 + 7 + 8 + 9 |
| 8 | 44 | 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 |
| 9 | 45 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 |
Using them at crossings
A single-combination run tells you which digits it contains, not their order. The order comes from the runs crossing it. Suppose a two-cell run of 16 (7 and 9) crosses a three-cell run of 23 (6, 8 and 9). The shared cell needs a digit that belongs to both sets, and 9 is the only one they have in common, so 9 goes there and the other cell of the 16 is 7.
Even a run with two or three combinations can help. Two cells adding up to 15 are 6+9 or 7+8, so they can never contain 1 to 5. A crossing run that needs a small digit at that cell is immediately limited.
Watch the edges of a field as well. A long run with a low total, such as 10 over four cells (1+2+3+4), floods its neighbourhood with small digits and makes the crossing runs much easier.
Practice
The easy kakuro fields here are built to be solved with combinations alone, so they are the best place to practise. Start with the easy set, then move on to free play at a larger size.