Cage combinations
Killer Sudoku cage combinations
Every possible distinct-digit combination for cages of 2–9 cells. Find a target total, note its candidate sets, then use the grid to determine their order.
Complete reference: 502 combinations across 120 valid size-and-sum rows.
Digits are always distinct and range from 1 to 9. Combinations are listed once, in ascending order; permutations are not repeated. Totals outside each section’s range are impossible.
Digits are always distinct and range from 1 to 9. Combinations are listed once, in ascending order; permutations are not repeated. Totals outside each section’s range are impossible.
A one-cell cage is simpler: its digit equals the printed sum (1–9). The playable game mostly uses 2–5-cell cages, while this reference also covers larger cages found in other Killer Sudoku puzzles.
2-cell cage combinations
Possible sums: 3–17. 36 distinct combinations.
| Sum | Sets | Digit combinations (any order) |
|---|---|---|
| 3 | 1 | 1 + 2 |
| 4 | 1 | 1 + 3 |
| 5 | 2 | 1 + 42 + 3 |
| 6 | 2 | 1 + 52 + 4 |
| 7 | 3 | 1 + 62 + 53 + 4 |
| 8 | 3 | 1 + 72 + 63 + 5 |
| 9 | 4 | 1 + 82 + 73 + 64 + 5 |
| 10 | 4 | 1 + 92 + 83 + 74 + 6 |
| 11 | 4 | 2 + 93 + 84 + 75 + 6 |
| 12 | 3 | 3 + 94 + 85 + 7 |
| 13 | 3 | 4 + 95 + 86 + 7 |
| 14 | 2 | 5 + 96 + 8 |
| 15 | 2 | 6 + 97 + 8 |
| 16 | 1 | 7 + 9 |
| 17 | 1 | 8 + 9 |
3-cell cage combinations
Possible sums: 6–24. 84 distinct combinations.
| Sum | Sets | Digit combinations (any order) |
|---|---|---|
| 6 | 1 | 1 + 2 + 3 |
| 7 | 1 | 1 + 2 + 4 |
| 8 | 2 | 1 + 2 + 51 + 3 + 4 |
| 9 | 3 | 1 + 2 + 61 + 3 + 52 + 3 + 4 |
| 10 | 4 | 1 + 2 + 71 + 3 + 61 + 4 + 52 + 3 + 5 |
| 11 | 5 | 1 + 2 + 81 + 3 + 71 + 4 + 62 + 3 + 62 + 4 + 5 |
| 12 | 7 | 1 + 2 + 91 + 3 + 81 + 4 + 71 + 5 + 62 + 3 + 72 + 4 + 63 + 4 + 5 |
| 13 | 7 | 1 + 3 + 91 + 4 + 81 + 5 + 72 + 3 + 82 + 4 + 72 + 5 + 63 + 4 + 6 |
| 14 | 8 | 1 + 4 + 91 + 5 + 81 + 6 + 72 + 3 + 92 + 4 + 82 + 5 + 73 + 4 + 73 + 5 + 6 |
| 15 | 8 | 1 + 5 + 91 + 6 + 82 + 4 + 92 + 5 + 82 + 6 + 73 + 4 + 83 + 5 + 74 + 5 + 6 |
| 16 | 8 | 1 + 6 + 91 + 7 + 82 + 5 + 92 + 6 + 83 + 4 + 93 + 5 + 83 + 6 + 74 + 5 + 7 |
| 17 | 7 | 1 + 7 + 92 + 6 + 92 + 7 + 83 + 5 + 93 + 6 + 84 + 5 + 84 + 6 + 7 |
| 18 | 7 | 1 + 8 + 92 + 7 + 93 + 6 + 93 + 7 + 84 + 5 + 94 + 6 + 85 + 6 + 7 |
| 19 | 5 | 2 + 8 + 93 + 7 + 94 + 6 + 94 + 7 + 85 + 6 + 8 |
| 20 | 4 | 3 + 8 + 94 + 7 + 95 + 6 + 95 + 7 + 8 |
| 21 | 3 | 4 + 8 + 95 + 7 + 96 + 7 + 8 |
| 22 | 2 | 5 + 8 + 96 + 7 + 9 |
| 23 | 1 | 6 + 8 + 9 |
| 24 | 1 | 7 + 8 + 9 |
4-cell cage combinations
Possible sums: 10–30. 126 distinct combinations.
| Sum | Sets | Digit combinations (any order) |
|---|---|---|
| 10 | 1 | 1 + 2 + 3 + 4 |
| 11 | 1 | 1 + 2 + 3 + 5 |
| 12 | 2 | 1 + 2 + 3 + 61 + 2 + 4 + 5 |
| 13 | 3 | 1 + 2 + 3 + 71 + 2 + 4 + 61 + 3 + 4 + 5 |
| 14 | 5 | 1 + 2 + 3 + 81 + 2 + 4 + 71 + 2 + 5 + 61 + 3 + 4 + 62 + 3 + 4 + 5 |
| 15 | 6 | 1 + 2 + 3 + 91 + 2 + 4 + 81 + 2 + 5 + 71 + 3 + 4 + 71 + 3 + 5 + 62 + 3 + 4 + 6 |
| 16 | 8 | 1 + 2 + 4 + 91 + 2 + 5 + 81 + 2 + 6 + 71 + 3 + 4 + 81 + 3 + 5 + 71 + 4 + 5 + 62 + 3 + 4 + 72 + 3 + 5 + 6 |
| 17 | 9 | 1 + 2 + 5 + 91 + 2 + 6 + 81 + 3 + 4 + 91 + 3 + 5 + 81 + 3 + 6 + 71 + 4 + 5 + 72 + 3 + 4 + 82 + 3 + 5 + 72 + 4 + 5 + 6 |
| 18 | 11 | 1 + 2 + 6 + 91 + 2 + 7 + 81 + 3 + 5 + 91 + 3 + 6 + 81 + 4 + 5 + 81 + 4 + 6 + 72 + 3 + 4 + 92 + 3 + 5 + 82 + 3 + 6 + 72 + 4 + 5 + 73 + 4 + 5 + 6 |
| 19 | 11 | 1 + 2 + 7 + 91 + 3 + 6 + 91 + 3 + 7 + 81 + 4 + 5 + 91 + 4 + 6 + 81 + 5 + 6 + 72 + 3 + 5 + 92 + 3 + 6 + 82 + 4 + 5 + 82 + 4 + 6 + 73 + 4 + 5 + 7 |
| 20 | 12 | 1 + 2 + 8 + 91 + 3 + 7 + 91 + 4 + 6 + 91 + 4 + 7 + 81 + 5 + 6 + 82 + 3 + 6 + 92 + 3 + 7 + 82 + 4 + 5 + 92 + 4 + 6 + 82 + 5 + 6 + 73 + 4 + 5 + 83 + 4 + 6 + 7 |
| 21 | 11 | 1 + 3 + 8 + 91 + 4 + 7 + 91 + 5 + 6 + 91 + 5 + 7 + 82 + 3 + 7 + 92 + 4 + 6 + 92 + 4 + 7 + 82 + 5 + 6 + 83 + 4 + 5 + 93 + 4 + 6 + 83 + 5 + 6 + 7 |
| 22 | 11 | 1 + 4 + 8 + 91 + 5 + 7 + 91 + 6 + 7 + 82 + 3 + 8 + 92 + 4 + 7 + 92 + 5 + 6 + 92 + 5 + 7 + 83 + 4 + 6 + 93 + 4 + 7 + 83 + 5 + 6 + 84 + 5 + 6 + 7 |
| 23 | 9 | 1 + 5 + 8 + 91 + 6 + 7 + 92 + 4 + 8 + 92 + 5 + 7 + 92 + 6 + 7 + 83 + 4 + 7 + 93 + 5 + 6 + 93 + 5 + 7 + 84 + 5 + 6 + 8 |
| 24 | 8 | 1 + 6 + 8 + 92 + 5 + 8 + 92 + 6 + 7 + 93 + 4 + 8 + 93 + 5 + 7 + 93 + 6 + 7 + 84 + 5 + 6 + 94 + 5 + 7 + 8 |
| 25 | 6 | 1 + 7 + 8 + 92 + 6 + 8 + 93 + 5 + 8 + 93 + 6 + 7 + 94 + 5 + 7 + 94 + 6 + 7 + 8 |
| 26 | 5 | 2 + 7 + 8 + 93 + 6 + 8 + 94 + 5 + 8 + 94 + 6 + 7 + 95 + 6 + 7 + 8 |
| 27 | 3 | 3 + 7 + 8 + 94 + 6 + 8 + 95 + 6 + 7 + 9 |
| 28 | 2 | 4 + 7 + 8 + 95 + 6 + 8 + 9 |
| 29 | 1 | 5 + 7 + 8 + 9 |
| 30 | 1 | 6 + 7 + 8 + 9 |
5-cell cage combinations
Possible sums: 15–35. 126 distinct combinations.
| Sum | Sets | Digit combinations (any order) |
|---|---|---|
| 15 | 1 | 1 + 2 + 3 + 4 + 5 |
| 16 | 1 | 1 + 2 + 3 + 4 + 6 |
| 17 | 2 | 1 + 2 + 3 + 4 + 71 + 2 + 3 + 5 + 6 |
| 18 | 3 | 1 + 2 + 3 + 4 + 81 + 2 + 3 + 5 + 71 + 2 + 4 + 5 + 6 |
| 19 | 5 | 1 + 2 + 3 + 4 + 91 + 2 + 3 + 5 + 81 + 2 + 3 + 6 + 71 + 2 + 4 + 5 + 71 + 3 + 4 + 5 + 6 |
| 20 | 6 | 1 + 2 + 3 + 5 + 91 + 2 + 3 + 6 + 81 + 2 + 4 + 5 + 81 + 2 + 4 + 6 + 71 + 3 + 4 + 5 + 72 + 3 + 4 + 5 + 6 |
| 21 | 8 | 1 + 2 + 3 + 6 + 91 + 2 + 3 + 7 + 81 + 2 + 4 + 5 + 91 + 2 + 4 + 6 + 81 + 2 + 5 + 6 + 71 + 3 + 4 + 5 + 81 + 3 + 4 + 6 + 72 + 3 + 4 + 5 + 7 |
| 22 | 9 | 1 + 2 + 3 + 7 + 91 + 2 + 4 + 6 + 91 + 2 + 4 + 7 + 81 + 2 + 5 + 6 + 81 + 3 + 4 + 5 + 91 + 3 + 4 + 6 + 81 + 3 + 5 + 6 + 72 + 3 + 4 + 5 + 82 + 3 + 4 + 6 + 7 |
| 23 | 11 | 1 + 2 + 3 + 8 + 91 + 2 + 4 + 7 + 91 + 2 + 5 + 6 + 91 + 2 + 5 + 7 + 81 + 3 + 4 + 6 + 91 + 3 + 4 + 7 + 81 + 3 + 5 + 6 + 81 + 4 + 5 + 6 + 72 + 3 + 4 + 5 + 92 + 3 + 4 + 6 + 82 + 3 + 5 + 6 + 7 |
| 24 | 11 | 1 + 2 + 4 + 8 + 91 + 2 + 5 + 7 + 91 + 2 + 6 + 7 + 81 + 3 + 4 + 7 + 91 + 3 + 5 + 6 + 91 + 3 + 5 + 7 + 81 + 4 + 5 + 6 + 82 + 3 + 4 + 6 + 92 + 3 + 4 + 7 + 82 + 3 + 5 + 6 + 82 + 4 + 5 + 6 + 7 |
| 25 | 12 | 1 + 2 + 5 + 8 + 91 + 2 + 6 + 7 + 91 + 3 + 4 + 8 + 91 + 3 + 5 + 7 + 91 + 3 + 6 + 7 + 81 + 4 + 5 + 6 + 91 + 4 + 5 + 7 + 82 + 3 + 4 + 7 + 92 + 3 + 5 + 6 + 92 + 3 + 5 + 7 + 82 + 4 + 5 + 6 + 83 + 4 + 5 + 6 + 7 |
| 26 | 11 | 1 + 2 + 6 + 8 + 91 + 3 + 5 + 8 + 91 + 3 + 6 + 7 + 91 + 4 + 5 + 7 + 91 + 4 + 6 + 7 + 82 + 3 + 4 + 8 + 92 + 3 + 5 + 7 + 92 + 3 + 6 + 7 + 82 + 4 + 5 + 6 + 92 + 4 + 5 + 7 + 83 + 4 + 5 + 6 + 8 |
| 27 | 11 | 1 + 2 + 7 + 8 + 91 + 3 + 6 + 8 + 91 + 4 + 5 + 8 + 91 + 4 + 6 + 7 + 91 + 5 + 6 + 7 + 82 + 3 + 5 + 8 + 92 + 3 + 6 + 7 + 92 + 4 + 5 + 7 + 92 + 4 + 6 + 7 + 83 + 4 + 5 + 6 + 93 + 4 + 5 + 7 + 8 |
| 28 | 9 | 1 + 3 + 7 + 8 + 91 + 4 + 6 + 8 + 91 + 5 + 6 + 7 + 92 + 3 + 6 + 8 + 92 + 4 + 5 + 8 + 92 + 4 + 6 + 7 + 92 + 5 + 6 + 7 + 83 + 4 + 5 + 7 + 93 + 4 + 6 + 7 + 8 |
| 29 | 8 | 1 + 4 + 7 + 8 + 91 + 5 + 6 + 8 + 92 + 3 + 7 + 8 + 92 + 4 + 6 + 8 + 92 + 5 + 6 + 7 + 93 + 4 + 5 + 8 + 93 + 4 + 6 + 7 + 93 + 5 + 6 + 7 + 8 |
| 30 | 6 | 1 + 5 + 7 + 8 + 92 + 4 + 7 + 8 + 92 + 5 + 6 + 8 + 93 + 4 + 6 + 8 + 93 + 5 + 6 + 7 + 94 + 5 + 6 + 7 + 8 |
| 31 | 5 | 1 + 6 + 7 + 8 + 92 + 5 + 7 + 8 + 93 + 4 + 7 + 8 + 93 + 5 + 6 + 8 + 94 + 5 + 6 + 7 + 9 |
| 32 | 3 | 2 + 6 + 7 + 8 + 93 + 5 + 7 + 8 + 94 + 5 + 6 + 8 + 9 |
| 33 | 2 | 3 + 6 + 7 + 8 + 94 + 5 + 7 + 8 + 9 |
| 34 | 1 | 4 + 6 + 7 + 8 + 9 |
| 35 | 1 | 5 + 6 + 7 + 8 + 9 |
6-cell cage combinations
Possible sums: 21–39. 84 distinct combinations.
| Sum | Sets | Digit combinations (any order) |
|---|---|---|
| 21 | 1 | 1 + 2 + 3 + 4 + 5 + 6 |
| 22 | 1 | 1 + 2 + 3 + 4 + 5 + 7 |
| 23 | 2 | 1 + 2 + 3 + 4 + 5 + 81 + 2 + 3 + 4 + 6 + 7 |
| 24 | 3 | 1 + 2 + 3 + 4 + 5 + 91 + 2 + 3 + 4 + 6 + 81 + 2 + 3 + 5 + 6 + 7 |
| 25 | 4 | 1 + 2 + 3 + 4 + 6 + 91 + 2 + 3 + 4 + 7 + 81 + 2 + 3 + 5 + 6 + 81 + 2 + 4 + 5 + 6 + 7 |
| 26 | 5 | 1 + 2 + 3 + 4 + 7 + 91 + 2 + 3 + 5 + 6 + 91 + 2 + 3 + 5 + 7 + 81 + 2 + 4 + 5 + 6 + 81 + 3 + 4 + 5 + 6 + 7 |
| 27 | 7 | 1 + 2 + 3 + 4 + 8 + 91 + 2 + 3 + 5 + 7 + 91 + 2 + 3 + 6 + 7 + 81 + 2 + 4 + 5 + 6 + 91 + 2 + 4 + 5 + 7 + 81 + 3 + 4 + 5 + 6 + 82 + 3 + 4 + 5 + 6 + 7 |
| 28 | 7 | 1 + 2 + 3 + 5 + 8 + 91 + 2 + 3 + 6 + 7 + 91 + 2 + 4 + 5 + 7 + 91 + 2 + 4 + 6 + 7 + 81 + 3 + 4 + 5 + 6 + 91 + 3 + 4 + 5 + 7 + 82 + 3 + 4 + 5 + 6 + 8 |
| 29 | 8 | 1 + 2 + 3 + 6 + 8 + 91 + 2 + 4 + 5 + 8 + 91 + 2 + 4 + 6 + 7 + 91 + 2 + 5 + 6 + 7 + 81 + 3 + 4 + 5 + 7 + 91 + 3 + 4 + 6 + 7 + 82 + 3 + 4 + 5 + 6 + 92 + 3 + 4 + 5 + 7 + 8 |
| 30 | 8 | 1 + 2 + 3 + 7 + 8 + 91 + 2 + 4 + 6 + 8 + 91 + 2 + 5 + 6 + 7 + 91 + 3 + 4 + 5 + 8 + 91 + 3 + 4 + 6 + 7 + 91 + 3 + 5 + 6 + 7 + 82 + 3 + 4 + 5 + 7 + 92 + 3 + 4 + 6 + 7 + 8 |
| 31 | 8 | 1 + 2 + 4 + 7 + 8 + 91 + 2 + 5 + 6 + 8 + 91 + 3 + 4 + 6 + 8 + 91 + 3 + 5 + 6 + 7 + 91 + 4 + 5 + 6 + 7 + 82 + 3 + 4 + 5 + 8 + 92 + 3 + 4 + 6 + 7 + 92 + 3 + 5 + 6 + 7 + 8 |
| 32 | 7 | 1 + 2 + 5 + 7 + 8 + 91 + 3 + 4 + 7 + 8 + 91 + 3 + 5 + 6 + 8 + 91 + 4 + 5 + 6 + 7 + 92 + 3 + 4 + 6 + 8 + 92 + 3 + 5 + 6 + 7 + 92 + 4 + 5 + 6 + 7 + 8 |
| 33 | 7 | 1 + 2 + 6 + 7 + 8 + 91 + 3 + 5 + 7 + 8 + 91 + 4 + 5 + 6 + 8 + 92 + 3 + 4 + 7 + 8 + 92 + 3 + 5 + 6 + 8 + 92 + 4 + 5 + 6 + 7 + 93 + 4 + 5 + 6 + 7 + 8 |
| 34 | 5 | 1 + 3 + 6 + 7 + 8 + 91 + 4 + 5 + 7 + 8 + 92 + 3 + 5 + 7 + 8 + 92 + 4 + 5 + 6 + 8 + 93 + 4 + 5 + 6 + 7 + 9 |
| 35 | 4 | 1 + 4 + 6 + 7 + 8 + 92 + 3 + 6 + 7 + 8 + 92 + 4 + 5 + 7 + 8 + 93 + 4 + 5 + 6 + 8 + 9 |
| 36 | 3 | 1 + 5 + 6 + 7 + 8 + 92 + 4 + 6 + 7 + 8 + 93 + 4 + 5 + 7 + 8 + 9 |
| 37 | 2 | 2 + 5 + 6 + 7 + 8 + 93 + 4 + 6 + 7 + 8 + 9 |
| 38 | 1 | 3 + 5 + 6 + 7 + 8 + 9 |
| 39 | 1 | 4 + 5 + 6 + 7 + 8 + 9 |
7-cell cage combinations
Possible sums: 28–42. 36 distinct combinations.
| Sum | Sets | Digit combinations (any order) |
|---|---|---|
| 28 | 1 | 1 + 2 + 3 + 4 + 5 + 6 + 7 |
| 29 | 1 | 1 + 2 + 3 + 4 + 5 + 6 + 8 |
| 30 | 2 | 1 + 2 + 3 + 4 + 5 + 6 + 91 + 2 + 3 + 4 + 5 + 7 + 8 |
| 31 | 2 | 1 + 2 + 3 + 4 + 5 + 7 + 91 + 2 + 3 + 4 + 6 + 7 + 8 |
| 32 | 3 | 1 + 2 + 3 + 4 + 5 + 8 + 91 + 2 + 3 + 4 + 6 + 7 + 91 + 2 + 3 + 5 + 6 + 7 + 8 |
| 33 | 3 | 1 + 2 + 3 + 4 + 6 + 8 + 91 + 2 + 3 + 5 + 6 + 7 + 91 + 2 + 4 + 5 + 6 + 7 + 8 |
| 34 | 4 | 1 + 2 + 3 + 4 + 7 + 8 + 91 + 2 + 3 + 5 + 6 + 8 + 91 + 2 + 4 + 5 + 6 + 7 + 91 + 3 + 4 + 5 + 6 + 7 + 8 |
| 35 | 4 | 1 + 2 + 3 + 5 + 7 + 8 + 91 + 2 + 4 + 5 + 6 + 8 + 91 + 3 + 4 + 5 + 6 + 7 + 92 + 3 + 4 + 5 + 6 + 7 + 8 |
| 36 | 4 | 1 + 2 + 3 + 6 + 7 + 8 + 91 + 2 + 4 + 5 + 7 + 8 + 91 + 3 + 4 + 5 + 6 + 8 + 92 + 3 + 4 + 5 + 6 + 7 + 9 |
| 37 | 3 | 1 + 2 + 4 + 6 + 7 + 8 + 91 + 3 + 4 + 5 + 7 + 8 + 92 + 3 + 4 + 5 + 6 + 8 + 9 |
| 38 | 3 | 1 + 2 + 5 + 6 + 7 + 8 + 91 + 3 + 4 + 6 + 7 + 8 + 92 + 3 + 4 + 5 + 7 + 8 + 9 |
| 39 | 2 | 1 + 3 + 5 + 6 + 7 + 8 + 92 + 3 + 4 + 6 + 7 + 8 + 9 |
| 40 | 2 | 1 + 4 + 5 + 6 + 7 + 8 + 92 + 3 + 5 + 6 + 7 + 8 + 9 |
| 41 | 1 | 2 + 4 + 5 + 6 + 7 + 8 + 9 |
| 42 | 1 | 3 + 4 + 5 + 6 + 7 + 8 + 9 |
8-cell cage combinations
Possible sums: 36–44. 9 distinct combinations.
| Sum | Sets | Digit combinations (any order) |
|---|---|---|
| 36 | 1 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 |
| 37 | 1 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 9 |
| 38 | 1 | 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 |
| 39 | 1 | 1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 |
| 40 | 1 | 1 + 2 + 3 + 4 + 6 + 7 + 8 + 9 |
| 41 | 1 | 1 + 2 + 3 + 5 + 6 + 7 + 8 + 9 |
| 42 | 1 | 1 + 2 + 4 + 5 + 6 + 7 + 8 + 9 |
| 43 | 1 | 1 + 3 + 4 + 5 + 6 + 7 + 8 + 9 |
| 44 | 1 | 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 |
9-cell cage combinations
Possible sums: 45–45. 1 distinct combinations.
| Sum | Sets | Digit combinations (any order) |
|---|---|---|
| 45 | 1 | 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 |
How to use the table
For a two-cell total of 10, the sets are 1 + 9, 2 + 8, 3 + 7 and 4 + 6. If a 9 is already in the same row as both cage cells, you can rule out 1 + 9. Continue checking the remaining candidates against Sudoku rules. See the rules and solving guide for more examples.
Frequently asked questions
Are the combinations ordered?
No. Each listed combination is an unordered set of distinct digits. The grid determines how those digits can be arranged in the cage.
Why are some totals not listed?
Only possible totals are listed. For n cells, the minimum is 1 + 2 + … + n and the maximum is 9 + 8 + … for n distinct digits. Totals outside that range are impossible.
Does one listed combination mean the cage is solved?
It fixes the set of digits, not their positions. You still need row, column, box and other cage restrictions to place each digit.