Killer Sudoku Combinations
Choose the size of a cage, type its sum and see every set of different digits that fits. Filter by digits you already know.
Choose the number of cells and type the sum to see the combinations here.
How to use the calculator
- Choose how many cells the cage has. The hint under Sum of the cage says which sums are possible.
- Type the sum. The combinations appear at once, with the digits that are in all of them and the digits that are in none.
- Under Must include, press digits that you know are in the cage. Under Exclude, press digits that cannot be in it, for example because the row or box already has them. Press a digit again to remove it. A digit can be in only one filter: pressing it in the other filter moves it there.
- Start over clears the sum and both filters and sets the cage back to 3 cells.
The rules behind the numbers: a cage holds different digits from 1 to 9, and its digits add up to the small number printed in its corner. The same holds for a run of white cells in Kakuro (also called cross sums), so the calculator and the tables work for both puzzles. Kakuro runs have 2 to 9 cells, so the 1-cell option does not apply there.
Example: a 3-cell cage with sum 15
Choose 3 cells and type 15. Eight combinations fit: 1 5 9, 1 6 8, 2 4 9, 2 5 8, 2 6 7, 3 4 8, 3 5 7 and 4 5 6. No digit is in all of them, and every digit is in at least one, so the cage alone tells you little.
Now suppose the cage lies in a row that already has a 5, and one of its cells can only be 9. Press 9 under Must include and 5 under Exclude: only 2 4 9 is left. The other two cells of the cage are 2 and 4, in some order.
The cages with a single combination are the best places to start. For 2 cells these are the sums 3, 4, 16 and 17; for 3 cells 6, 7, 23 and 24; for 4 cells 10, 11, 29 and 30.
How the combinations are counted
The calculator goes through all 511 sets of different digits from 1 to 9 and keeps those with the chosen size, sum and filters. The number of sets of n digits is C(9, n), “9 choose n”: 9 sets of one digit, 36 of two, 84 of three, 126 of four, 126 of five, 84 of six, 36 of seven, 9 of eight and 1 of nine.
The smallest sum of n different digits is 1 + 2 + … + n, and the largest is 9 + 8 + … + (10 − n). For 3 cells that is 6 to 24.
A useful mirror: the digits 1 to 9 add up to 45, so each combination of n digits with sum s leaves out 9 − n digits with sum 45 − s. The two lists match one to one. A 7-cell cage with sum 40 leaves out two digits that add up to 5: 1 and 4, or 2 and 3.
Cheat sheet: 2 to 5 cells
Each table lists every sum that is possible for the size, how many combinations it has, and the combinations themselves. A cage of 1 cell holds its sum; a cage of 9 cells holds every digit and adds up to 45.
| Sum | Number | 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 |
| Sum | Number | 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 |
| Sum | Number | 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 |
| Sum | Number | 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 |
Cheat sheet: 6 to 8 cells
| Sum | Number | Combinations |
|---|---|---|
| 21 | 1 | 1 2 3 4 5 6 |
| 22 | 1 | 1 2 3 4 5 7 |
| 23 | 2 | 1 2 3 4 5 8, 1 2 3 4 6 7 |
| 24 | 3 | 1 2 3 4 5 9, 1 2 3 4 6 8, 1 2 3 5 6 7 |
| 25 | 4 | 1 2 3 4 6 9, 1 2 3 4 7 8, 1 2 3 5 6 8, 1 2 4 5 6 7 |
| 26 | 5 | 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 |
| 27 | 7 | 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 |
| 28 | 7 | 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 |
| 29 | 8 | 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 |
| 30 | 8 | 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 |
| 31 | 8 | 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 |
| 32 | 7 | 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 |
| 33 | 7 | 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 |
| 34 | 5 | 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 |
| 35 | 4 | 1 4 6 7 8 9, 2 3 6 7 8 9, 2 4 5 7 8 9, 3 4 5 6 8 9 |
| 36 | 3 | 1 5 6 7 8 9, 2 4 6 7 8 9, 3 4 5 7 8 9 |
| 37 | 2 | 2 5 6 7 8 9, 3 4 6 7 8 9 |
| 38 | 1 | 3 5 6 7 8 9 |
| 39 | 1 | 4 5 6 7 8 9 |
| Sum | Number | Combinations |
|---|---|---|
| 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 9, 1 2 3 4 5 7 8 |
| 31 | 2 | 1 2 3 4 5 7 9, 1 2 3 4 6 7 8 |
| 32 | 3 | 1 2 3 4 5 8 9, 1 2 3 4 6 7 9, 1 2 3 5 6 7 8 |
| 33 | 3 | 1 2 3 4 6 8 9, 1 2 3 5 6 7 9, 1 2 4 5 6 7 8 |
| 34 | 4 | 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 |
| 35 | 4 | 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 |
| 36 | 4 | 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 |
| 37 | 3 | 1 2 4 6 7 8 9, 1 3 4 5 7 8 9, 2 3 4 5 6 8 9 |
| 38 | 3 | 1 2 5 6 7 8 9, 1 3 4 6 7 8 9, 2 3 4 5 7 8 9 |
| 39 | 2 | 1 3 5 6 7 8 9, 2 3 4 6 7 8 9 |
| 40 | 2 | 1 4 5 6 7 8 9, 2 3 5 6 7 8 9 |
| 41 | 1 | 2 4 5 6 7 8 9 |
| 42 | 1 | 3 4 5 6 7 8 9 |
| Sum | Number | Combinations |
|---|---|---|
| 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 |
Frequently asked questions
- Can a killer sudoku cage repeat a digit?
- No. The digits in a cage are all different, which is why each combination here uses a digit at most once. A cage can still hold the same digits as a neighbouring cage.
- Which cages have only one combination?
- For 2 to 7 cells, the two smallest and the two largest sums, such as 3, 4, 16 and 17 for two cells. For 8 cells every sum has one combination, because the cage leaves out a single digit.
- Does the cheat sheet work for Kakuro?
- Yes. A Kakuro run also holds different digits from 1 to 9 that add up to its clue, so the same combinations apply.
- How do I use the 45 rule?
- Every row, column and box of a killer sudoku adds up to 45. If cages that lie entirely inside a box cover all of it except one cell, that cell is 45 minus the sum of those cages.
Last updated: September 29, 2026