Killer sudoku combinations
Every combination of different digits for every cage size and sum, plus a lookup that filters by digits you already know.
Quick lookup
8 combinations for 3 cells adding up to 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
How to use cage combinations
In killer sudoku, the grid is divided into cages, each with a small number in its corner: the sum of the digits inside. Digits in a cage are all different. Knowing which sets of digits can make a given sum is the single most useful skill in killer sudoku, because many cages have very few options. A two-cell cage of 3 can only be 1 and 2, so those two cells hold exactly 1 and 2 in some order, and 1 and 2 can be removed from the rest of any row, column or box that contains the whole cage.
The table below lists every combination for every cage size from 2 to 9 cells. 34 of the 120 size-and-sum pairs have exactly one combination; they are marked “only one” and are the best places to start a killer puzzle. The lookup above does the same job interactively: choose the cage size and sum, then tick digits you already know are in the cage, or that cannot be, to narrow the list.
Patterns worth memorising
- The extremes are fixed. For any cage size, the smallest and largest possible sums have one combination: 2 cells: 3 = 1+2 and 17 = 8+9; 3 cells: 6 = 1+2+3 and 24 = 7+8+9; 4 cells: 10 = 1+2+3+4 and 30 = 6+7+8+9.
- One step in from the extremes is also fixed. 2 cells: 4 = 1+3 and 16 = 7+9; 3 cells: 7 = 1+2+4 and 23 = 6+8+9.
- Large cages mirror small ones. All nine digits add up to 45, so an 8-cell cage is just “45 minus the missing digit”, and a 7-cell cage of 45 − x is the complement of a 2-cell cage of x.
- Every row, column and box adds up to 45. This “rule of 45” lets you work out a single cell that sticks out of a group of cages in a box or row.
Combinations tell you which digits a cage contains, not where they go. Combine them with ordinary sudoku logic: a digit that appears in every combination for a cage must be somewhere in that cage, and a digit that appears in none can be removed from all its cells.
Full combination table
2 cells (sums 3–17)
| Sum | Combinations |
|---|---|
| 3 | 12 only one |
| 4 | 13 only one |
| 5 | 14 23 |
| 6 | 15 24 |
| 7 | 16 25 34 |
| 8 | 17 26 35 |
| 9 | 18 27 36 45 |
| 10 | 19 28 37 46 |
| 11 | 29 38 47 56 |
| 12 | 39 48 57 |
| 13 | 49 58 67 |
| 14 | 59 68 |
| 15 | 69 78 |
| 16 | 79 only one |
| 17 | 89 only one |
3 cells (sums 6–24)
| Sum | Combinations |
|---|---|
| 6 | 123 only one |
| 7 | 124 only one |
| 8 | 125 134 |
| 9 | 126 135 234 |
| 10 | 127 136 145 235 |
| 11 | 128 137 146 236 245 |
| 12 | 129 138 147 156 237 246 345 |
| 13 | 139 148 157 238 247 256 346 |
| 14 | 149 158 167 239 248 257 347 356 |
| 15 | 159 168 249 258 267 348 357 456 |
| 16 | 169 178 259 268 349 358 367 457 |
| 17 | 179 269 278 359 368 458 467 |
| 18 | 189 279 369 378 459 468 567 |
| 19 | 289 379 469 478 568 |
| 20 | 389 479 569 578 |
| 21 | 489 579 678 |
| 22 | 589 679 |
| 23 | 689 only one |
| 24 | 789 only one |
4 cells (sums 10–30)
| Sum | Combinations |
|---|---|
| 10 | 1234 only one |
| 11 | 1235 only one |
| 12 | 1236 1245 |
| 13 | 1237 1246 1345 |
| 14 | 1238 1247 1256 1346 2345 |
| 15 | 1239 1248 1257 1347 1356 2346 |
| 16 | 1249 1258 1267 1348 1357 1456 2347 2356 |
| 17 | 1259 1268 1349 1358 1367 1457 2348 2357 2456 |
| 18 | 1269 1278 1359 1368 1458 1467 2349 2358 2367 2457 3456 |
| 19 | 1279 1369 1378 1459 1468 1567 2359 2368 2458 2467 3457 |
| 20 | 1289 1379 1469 1478 1568 2369 2378 2459 2468 2567 3458 3467 |
| 21 | 1389 1479 1569 1578 2379 2469 2478 2568 3459 3468 3567 |
| 22 | 1489 1579 1678 2389 2479 2569 2578 3469 3478 3568 4567 |
| 23 | 1589 1679 2489 2579 2678 3479 3569 3578 4568 |
| 24 | 1689 2589 2679 3489 3579 3678 4569 4578 |
| 25 | 1789 2689 3589 3679 4579 4678 |
| 26 | 2789 3689 4589 4679 5678 |
| 27 | 3789 4689 5679 |
| 28 | 4789 5689 |
| 29 | 5789 only one |
| 30 | 6789 only one |
5 cells (sums 15–35)
| Sum | Combinations |
|---|---|
| 15 | 12345 only one |
| 16 | 12346 only one |
| 17 | 12347 12356 |
| 18 | 12348 12357 12456 |
| 19 | 12349 12358 12367 12457 13456 |
| 20 | 12359 12368 12458 12467 13457 23456 |
| 21 | 12369 12378 12459 12468 12567 13458 13467 23457 |
| 22 | 12379 12469 12478 12568 13459 13468 13567 23458 23467 |
| 23 | 12389 12479 12569 12578 13469 13478 13568 14567 23459 23468 23567 |
| 24 | 12489 12579 12678 13479 13569 13578 14568 23469 23478 23568 24567 |
| 25 | 12589 12679 13489 13579 13678 14569 14578 23479 23569 23578 24568 34567 |
| 26 | 12689 13589 13679 14579 14678 23489 23579 23678 24569 24578 34568 |
| 27 | 12789 13689 14589 14679 15678 23589 23679 24579 24678 34569 34578 |
| 28 | 13789 14689 15679 23689 24589 24679 25678 34579 34678 |
| 29 | 14789 15689 23789 24689 25679 34589 34679 35678 |
| 30 | 15789 24789 25689 34689 35679 45678 |
| 31 | 16789 25789 34789 35689 45679 |
| 32 | 26789 35789 45689 |
| 33 | 36789 45789 |
| 34 | 46789 only one |
| 35 | 56789 only one |
6 cells (sums 21–39)
| Sum | Combinations |
|---|---|
| 21 | 123456 only one |
| 22 | 123457 only one |
| 23 | 123458 123467 |
| 24 | 123459 123468 123567 |
| 25 | 123469 123478 123568 124567 |
| 26 | 123479 123569 123578 124568 134567 |
| 27 | 123489 123579 123678 124569 124578 134568 234567 |
| 28 | 123589 123679 124579 124678 134569 134578 234568 |
| 29 | 123689 124589 124679 125678 134579 134678 234569 234578 |
| 30 | 123789 124689 125679 134589 134679 135678 234579 234678 |
| 31 | 124789 125689 134689 135679 145678 234589 234679 235678 |
| 32 | 125789 134789 135689 145679 234689 235679 245678 |
| 33 | 126789 135789 145689 234789 235689 245679 345678 |
| 34 | 136789 145789 235789 245689 345679 |
| 35 | 146789 236789 245789 345689 |
| 36 | 156789 246789 345789 |
| 37 | 256789 346789 |
| 38 | 356789 only one |
| 39 | 456789 only one |
7 cells (sums 28–42)
| Sum | Combinations |
|---|---|
| 28 | 1234567 only one |
| 29 | 1234568 only one |
| 30 | 1234569 1234578 |
| 31 | 1234579 1234678 |
| 32 | 1234589 1234679 1235678 |
| 33 | 1234689 1235679 1245678 |
| 34 | 1234789 1235689 1245679 1345678 |
| 35 | 1235789 1245689 1345679 2345678 |
| 36 | 1236789 1245789 1345689 2345679 |
| 37 | 1246789 1345789 2345689 |
| 38 | 1256789 1346789 2345789 |
| 39 | 1356789 2346789 |
| 40 | 1456789 2356789 |
| 41 | 2456789 only one |
| 42 | 3456789 only one |
8 cells (sums 36–44)
| Sum | Combinations |
|---|---|
| 36 | 12345678 only one |
| 37 | 12345679 only one |
| 38 | 12345689 only one |
| 39 | 12345789 only one |
| 40 | 12346789 only one |
| 41 | 12356789 only one |
| 42 | 12456789 only one |
| 43 | 13456789 only one |
| 44 | 23456789 only one |
9 cells (sums 45–45)
| Sum | Combinations |
|---|---|
| 45 | 123456789 only one |
Frequently asked questions
What is a cage combination?
In killer sudoku each dashed cage shows a sum, and the digits in a cage are all different. A combination is a set of different digits 1-9 with the right number of digits that adds up to that sum.
Which cages have only one combination?
Many small and large sums are fixed. Two cells adding up to 3 must be 1 and 2; two cells adding up to 17 must be 8 and 9; three cells adding up to 6 must be 1, 2 and 3; nine cells always add up to 45.
What is the smallest and largest sum for a cage?
For n cells the smallest sum is 1+2+…+n and the largest is 9+8+… down n digits. For example, 3 cells range from 6 to 24.
Can a digit repeat inside a cage?
Not in standard killer sudoku. All digits in a cage are different, which is what this table assumes.
How do I use the include and exclude boxes?
Tick digits you already know are in the cage under “Must include” and digits that cannot be in it under “Must not include”; the list shrinks to the combinations that still fit.
Can I print this table?
Yes. Use your browser’s print command; the table prints without the site header and ads.
Updated · Written by the NineByNine team