Killer Sudoku Combinations
Killer sudoku combinations are the sets of digits that can fill a cage of a given size and add up to its target - also called cage combinations, cage sums, or simply the killer cheat sheet. Every cage in Killer Sudoku shows a target sum, and the first question is always the same: which digits can add up to that? This page is the complete answer: every combination for every cage size from 2 to 9 cells, all 502 of them, with the sums that have only one answer in bold. Keep it open while you play.
What is Killer Sudoku?
Killer Sudoku is Classic Sudoku with one extra layer. The usual rules still hold - each row, column and 3×3 box contains 1 to 9 exactly once - but the grid is also divided into dotted-outline cages, each printed with a small target number. The digits in a cage must add up to that target, and a digit never repeats inside a cage. That last rule is what makes the tables below finite and useful: a cage is a set of distinct digits, so a 3-cell cage summing to 24 can only ever be 7+8+9.
Killer grids usually start with no given digits at all. The cages are the clues.
How to use the tables
Find your cage's size (how many cells it covers), then its target sum. The row lists every set of digits that can fill it. Combinations are written compactly - 137 means the digits 1, 3 and 7 in some order.
If only one combination is listed, you have the cage's exact digits outright, and those are the rows worth memorising. If several are listed, you have still narrowed things enormously, and the rest of the grid will pick the winner: cross out any combination containing a digit that is already used in the cage's row, column or box.
Full combination tables
Cages of 2 to 9 cells. A 1-cell cage needs no table - its sum is its digit.
2-cell cages
Sums 3 to 17 · 36 combinations in total. Bold rows are the sums with only one possible combination.
| Sum | Possible digits |
|---|---|
| 3 | 12 |
| 4 | 13 |
| 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 |
| 17 | 89 |
3-cell cages
Sums 6 to 24 · 84 combinations in total. Bold rows are the sums with only one possible combination.
| Sum | Possible digits |
|---|---|
| 6 | 123 |
| 7 | 124 |
| 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 |
| 24 | 789 |
4-cell cages
Sums 10 to 30 · 126 combinations in total. Bold rows are the sums with only one possible combination.
| Sum | Possible digits |
|---|---|
| 10 | 1234 |
| 11 | 1235 |
| 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 |
| 30 | 6789 |
5-cell cages
Sums 15 to 35 · 126 combinations in total. Bold rows are the sums with only one possible combination.
| Sum | Possible digits |
|---|---|
| 15 | 12345 |
| 16 | 12346 |
| 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 |
| 35 | 56789 |
6-cell cages
Sums 21 to 39 · 84 combinations in total. Bold rows are the sums with only one possible combination.
| Sum | Possible digits |
|---|---|
| 21 | 123456 |
| 22 | 123457 |
| 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 |
| 39 | 456789 |
7-cell cages
Sums 28 to 42 · 36 combinations in total. Bold rows are the sums with only one possible combination.
| Sum | Possible digits |
|---|---|
| 28 | 1234567 |
| 29 | 1234568 |
| 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 |
| 42 | 3456789 |
8-cell cages
Sums 36 to 44 · 9 combinations in total. Bold rows are the sums with only one possible combination.
| Sum | Possible digits |
|---|---|
| 36 | 12345678 |
| 37 | 12345679 |
| 38 | 12345689 |
| 39 | 12345789 |
| 40 | 12346789 |
| 41 | 12356789 |
| 42 | 12456789 |
| 43 | 13456789 |
| 44 | 23456789 |
9-cell cages
Sums 45 to 45 · 1 combinations in total. Bold rows are the sums with only one possible combination.
| Sum | Possible digits |
|---|---|
| 45 | 123456789 |
The cheat sheet: sums with one and only one combination
These sums have exactly one combination, so a cage of that size and sum tells you its digits with certainty - no deduction needed. They are the highest-value rows on this page:
| Cells | Sum → digits |
|---|---|
| 2 | 3 → 12 · 4 → 13 · 16 → 79 · 17 → 89 |
| 3 | 6 → 123 · 7 → 124 · 23 → 689 · 24 → 789 |
| 4 | 10 → 1234 · 11 → 1235 · 29 → 5789 · 30 → 6789 |
| 5 | 15 → 12345 · 16 → 12346 · 34 → 46789 · 35 → 56789 |
| 6 | 21 → 123456 · 22 → 123457 · 38 → 356789 · 39 → 456789 |
| 7 | 28 → 1234567 · 29 → 1234568 · 41 → 2456789 · 42 → 3456789 |
| 8 | 36 → 12345678 · 37 → 12345679 · 38 → 12345689 · 39 → 12345789 · 40 → 12346789 · 41 → 12356789 · 42 → 12456789 · 43 → 13456789 · 44 → 23456789 |
| 9 | 45 → 123456789 |
The pattern is worth noticing: the unique combinations are always the lowest and highest sums a cage can take, because there is only one way to be that small or that large. Eight-cell cages are unique at every sum - the cage holds all but one digit, and the missing digit is fixed by the total (all nine digits sum to 45, so an 8-cell cage summing to 40 is missing the 5).
Putting them to work
Combinations rarely finish a cage on their own - they hand you a short list, and the grid narrows it. A two-cell cage summing to 16 is 79 for certain. A three-cell 8 is either 125 or 134, and if a 4 already sits in that row, it must be 125.
Two habits make the tables pay off fastest. First, scan a new grid for the unique cages above and pencil them in immediately - they are free information. Second, watch for cages trapped inside a single row, column or box: those digits are then excluded from the rest of that unit, which often cracks a neighbouring cage straight away.
Pair this reference with the rule of 45 and innies and outies, and most Killer grids open up. When a cage spans two boxes, cage splitting and cage-unit overlap take over.
Ready to use them? Play Killer Sudoku and keep this page open in a second tab.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.
Where to practise this
This technique starts paying off at Easy, which is where puzzles first stop yielding to anything simpler. Reading about a pattern and spotting it on a live grid are different skills - the second one only comes from playing.
Play Easy Sudoku About Easy Sudoku