Killer sudoku combinations: the complete cheat sheet
Every possible way to fill a cage, for every cage size and sum — with the forced combinations highlighted. Bookmark this page; it does half the work of any killer sudoku for you.
In killer sudoku, a cage's sum limits which digits can live inside it. Some sums are generous — a 3-cell cage totalling 15 can be filled eight different ways. Others are a gift: a 2-cell cage totalling 17 can only be 8+9. Learning to spot the forced ones is the single biggest upgrade you can make to your solving.
The forced combinations (memorise these)
These sums have exactly one possible combination. When you see one, you immediately know every digit in the cage — you just don't know the order yet.
| Cage size | Sum | Only combination |
|---|---|---|
| 2 cells | 3 | 1+2 |
| 2 cells | 4 | 1+3 |
| 2 cells | 16 | 7+9 |
| 2 cells | 17 | 8+9 |
| 3 cells | 6 | 1+2+3 |
| 3 cells | 7 | 1+2+4 |
| 3 cells | 23 | 6+8+9 |
| 3 cells | 24 | 7+8+9 |
| 4 cells | 10 | 1+2+3+4 |
| 4 cells | 11 | 1+2+3+5 |
| 4 cells | 29 | 5+7+8+9 |
| 4 cells | 30 | 6+7+8+9 |
| 5 cells | 15 | 1+2+3+4+5 |
| 5 cells | 16 | 1+2+3+4+6 |
| 5 cells | 34 | 4+6+7+8+9 |
| 5 cells | 35 | 5+6+7+8+9 |
2-cell cages
Sums run from 3 (1+2) to 17 (8+9). Note that a 2-cell cage can never sum to 10 with a repeated digit — 5+5 is illegal — so 10 has only four options, not five.
| Sum | Ways | Combinations |
|---|---|---|
| 3 | 1 | 1+2 |
| 4 | 1 | 1+3 |
| 5 | 2 | 1+4 · 2+3 |
| 6 | 2 | 1+5 · 2+4 |
| 7 | 3 | 1+6 · 2+5 · 3+4 |
| 8 | 3 | 1+7 · 2+6 · 3+5 |
| 9 | 4 | 1+8 · 2+7 · 3+6 · 4+5 |
| 10 | 4 | 1+9 · 2+8 · 3+7 · 4+6 |
| 11 | 4 | 2+9 · 3+8 · 4+7 · 5+6 |
| 12 | 3 | 3+9 · 4+8 · 5+7 |
| 13 | 3 | 4+9 · 5+8 · 6+7 |
| 14 | 2 | 5+9 · 6+8 |
| 15 | 2 | 6+9 · 7+8 |
| 16 | 1 | 7+9 |
| 17 | 1 | 8+9 |
3-cell cages
Sums run from 6 to 24. The middle sums (13–17) are the murkiest, with up to eight combinations each; lean on row and box constraints to narrow them.
| Sum | Ways | Combinations |
|---|---|---|
| 6 | 1 | 1+2+3 |
| 7 | 1 | 1+2+4 |
| 8 | 2 | 1+2+5 · 1+3+4 |
| 9 | 3 | 1+2+6 · 1+3+5 · 2+3+4 |
| 10 | 4 | 1+2+7 · 1+3+6 · 1+4+5 · 2+3+5 |
| 11 | 5 | 1+2+8 · 1+3+7 · 1+4+6 · 2+3+6 · 2+4+5 |
| 12 | 7 | 1+2+9 · 1+3+8 · 1+4+7 · 1+5+6 · 2+3+7 · 2+4+6 · 3+4+5 |
| 13 | 7 | 1+3+9 · 1+4+8 · 1+5+7 · 2+3+8 · 2+4+7 · 2+5+6 · 3+4+6 |
| 14 | 8 | 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 | 8 | 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 | 8 | 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 | 7 | 1+7+9 · 2+6+9 · 2+7+8 · 3+5+9 · 3+6+8 · 4+5+8 · 4+6+7 |
| 18 | 7 | 1+8+9 · 2+7+9 · 3+6+9 · 3+7+8 · 4+5+9 · 4+6+8 · 5+6+7 |
| 19 | 5 | 2+8+9 · 3+7+9 · 4+6+9 · 4+7+8 · 5+6+8 |
| 20 | 4 | 3+8+9 · 4+7+9 · 5+6+9 · 5+7+8 |
| 21 | 3 | 4+8+9 · 5+7+9 · 6+7+8 |
| 22 | 2 | 5+8+9 · 6+7+9 |
| 23 | 1 | 6+8+9 |
| 24 | 1 | 7+8+9 |
4-cell cages
Sums run from 10 to 30. With four cells the combination count balloons in the middle, but the digits excluded become the useful information: a 4-cell cage summing 12 can never contain a 9, for example.
| Sum | Ways | Combinations |
|---|---|---|
| 10 | 1 | 1+2+3+4 |
| 11 | 1 | 1+2+3+5 |
| 12 | 2 | 1+2+3+6 · 1+2+4+5 |
| 13 | 3 | 1+2+3+7 · 1+2+4+6 · 1+3+4+5 |
| 14 | 5 | 1+2+3+8 · 1+2+4+7 · 1+2+5+6 · 1+3+4+6 · 2+3+4+5 |
| 15 | 6 | 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 | 8 | 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 | 9 | 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 | 11 | 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 | 11 | 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 | 12 | 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 | 11 | 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 | 11 | 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 | 9 | 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 | 8 | 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 | 6 | 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 | 5 | 2+7+8+9 · 3+6+8+9 · 4+5+8+9 · 4+6+7+9 · 5+6+7+8 |
| 27 | 3 | 3+7+8+9 · 4+6+8+9 · 5+6+7+9 |
| 28 | 2 | 4+7+8+9 · 5+6+8+9 |
| 29 | 1 | 5+7+8+9 |
| 30 | 1 | 6+7+8+9 |
5-cell cages
Sums run from 15 to 35. At this size it's often easier to think about which four digits are missing from the cage: a 5-cell cage summing 35 is missing digits totalling 10.
| Sum | Ways | Combinations |
|---|---|---|
| 15 | 1 | 1+2+3+4+5 |
| 16 | 1 | 1+2+3+4+6 |
| 17 | 2 | 1+2+3+4+7 · 1+2+3+5+6 |
| 18 | 3 | 1+2+3+4+8 · 1+2+3+5+7 · 1+2+4+5+6 |
| 19 | 5 | 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 | 6 | 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 | 8 | 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 | 9 | 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 | 11 | 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 | 11 | 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 | 12 | 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 | 11 | 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 | 11 | 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 | 9 | 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 | 8 | 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 | 6 | 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 | 5 | 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 | 3 | 2+6+7+8+9 · 3+5+7+8+9 · 4+5+6+8+9 |
| 33 | 2 | 3+6+7+8+9 · 4+5+7+8+9 |
| 34 | 1 | 4+6+7+8+9 |
| 35 | 1 | 5+6+7+8+9 |
How to actually use these tables
- Start at the extremes. Scan the puzzle for any cage whose sum appears in the forced table above. Pencil-mark those digits immediately.
- Intersect with sudoku rules. A 2-cell cage summing 16 is 7+9 — but if a 7 already sits in that row, the cage cell in that row must be the 9.
- Use exclusions. Even a sum with many combinations excludes digits. A 2-cell cage summing 5 (1+4 or 2+3) can never contain 5, 6, 7, 8 or 9 — that's five eliminations for free.
- Combine with the rule of 45. Cage combinations plus the rule of 45 will carry you through most Hard puzzles without guessing.