Sudoku Written by Brian Hamilton and Karan Hamilton.
Brian and Karan are the puzzle lovers and game makers behind Sudoku Online Puzzles. You can learn more on the About page.
Guides are reviewed for clarity, accuracy, and usefulness for solvers. See our editorial standards, puzzle testing process, or send a correction.
A killer cage gives you two numbers before you know anything else about the board: how many cells it has and what they add up to. Together they fix which sets of different digits can fill it. Three cells making 22 hold 5, 8 and 9 or 6, 7 and 9, and nothing else.
This page lists every such set for every cage size from two cells to nine: 502 sets across 120 totals. We generated the chart with a short program rather than typing it, and the same program checks the tables on this page against its own output, so a typo can’t creep in. Below the chart, two real boards from our Killer Sudoku game show what the chart is for, and our own boards show which totals you will actually meet.
A combination belongs to the whole cage: 589 says the cage holds a 5, an 8 and a 9, in some order. A candidate belongs to one cell. The chart narrows the first, and the rows, columns and boxes sort out the second. Never played a killer? Our guide to the rules of Killer Sudoku walks through a first board.
Which killer sudoku totals have only one combination?
These are the cages that give their digits away on sight. For every size from two cells to seven, it is exactly the two smallest and the two largest totals. An eight-cell cage leaves out one digit, so every total it can have has one set, and a nine-cell cage always makes 45. That is 34 of the 120 totals.
| Cells | Totals with one set | The sets |
|---|---|---|
| 2 | 3, 4, 16, 17 | 12 13 79 89 |
| 3 | 6, 7, 23, 24 | 123 124 689 789 |
| 4 | 10, 11, 29, 30 | 1234 1235 5789 6789 |
| 5 | 15, 16, 34, 35 | 12345 12346 46789 56789 |
| 6 | 21, 22, 38, 39 | 123456 123457 356789 456789 |
| 7 | 28, 29, 41, 42 | 1234567 1234568 2456789 3456789 |
| 8 | every total, 36 to 44 | 1 to 9 without 45 − total |
| 9 | 45 | 123456789 |
Every total for each cage size, enumerated by program. Sets are written as their digits: 1234 means 1, 2, 3 and 4.
The pattern has a simple cause. The smallest total for three cells is 1 + 2 + 3 = 6. One more, 7, can only come from raising the largest digit: 1 + 2 + 4. Two more, 8, already has two ways, 1 + 2 + 5 and 1 + 3 + 4, and from there the choices multiply until the middle of the range: three cells making 14, 15 or 16 can be filled eight ways, and four or five cells peak at twelve, at totals of 20 and 25.
2-cell and 3-cell killer sudoku combinations
Two columns make the chart more than a list. In every set is a digit the cage is sure to hold, whichever set it turns out to be. In no set is a digit no cell of the cage can take. Every two-cell total rules something out: a 10 never holds a 5, since that would need 5 + 5, and an 11 never holds a 1, since its partner would have to be 10.
| Total | Sets | In every set | In no set |
|---|---|---|---|
| 3 | 12 | 12 | 3456789 |
| 4 | 13 | 13 | 2456789 |
| 5 | 14 23 | — | 56789 |
| 6 | 15 24 | — | 36789 |
| 7 | 16 25 34 | — | 789 |
| 8 | 17 26 35 | — | 489 |
| 9 | 18 27 36 45 | — | 9 |
| 10 | 19 28 37 46 | — | 5 |
| 11 | 29 38 47 56 | — | 1 |
| 12 | 39 48 57 | — | 126 |
| 13 | 49 58 67 | — | 123 |
| 14 | 59 68 | — | 12347 |
| 15 | 69 78 | — | 12345 |
| 16 | 79 | 79 | 1234568 |
| 17 | 89 | 89 | 1234567 |
| Total | Sets | In every set | In no set |
|---|---|---|---|
| 6 | 123 | 123 | 456789 |
| 7 | 124 | 124 | 356789 |
| 8 | 125 134 | 1 | 6789 |
| 9 | 126 135 234 | — | 789 |
| 10 | 127 136 145 235 | — | 89 |
| 11 | 128 137 146 236 245 | — | 9 |
| 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 | — | 1 |
| 20 | 389 479 569 578 | — | 12 |
| 21 | 489 579 678 | — | 123 |
| 22 | 589 679 | 9 | 1234 |
| 23 | 689 | 689 | 123457 |
| 24 | 789 | 789 | 123456 |
Highlighted rows have a single set. Each set is written as its digits in increasing order: 127 means 1, 2 and 7.
Three-cell totals are where the middle columns start to earn their place. Only 12 of the 19 rule any digit out, and among the totals with more than one set only two share a digit across all of their sets: every 8 holds a 1, and every 22 holds a 9.
4-cell and 5-cell killer sudoku combinations
Four- and five-cell cages are the big ones on our Hard and Expert boards, and the long lists hide some useful certainties. Four cells making 12 can be 1236 or 1245, so the cage always holds a 1 and a 2 and never a 7, 8 or 9, whichever set it is.
| Total | Sets | In every set | In no set |
|---|---|---|---|
| 10 | 1234 | 1234 | 56789 |
| 11 | 1235 | 1235 | 46789 |
| 12 | 1236 1245 | 12 | 789 |
| 13 | 1237 1246 1345 | 1 | 89 |
| 14 | 1238 1247 1256 1346 2345 | — | 9 |
| 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 | — | 1 |
| 27 | 3789 4689 5679 | 9 | 12 |
| 28 | 4789 5689 | 89 | 123 |
| 29 | 5789 | 5789 | 12346 |
| 30 | 6789 | 6789 | 12345 |
| Total | Sets | In every set | In no set |
|---|---|---|---|
| 15 | 12345 | 12345 | 6789 |
| 16 | 12346 | 12346 | 5789 |
| 17 | 12347 12356 | 123 | 89 |
| 18 | 12348 12357 12456 | 12 | 9 |
| 19 | 12349 12358 12367 12457 13456 | 1 | — |
| 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 | 9 | — |
| 32 | 26789 35789 45689 | 89 | 1 |
| 33 | 36789 45789 | 789 | 12 |
| 34 | 46789 | 46789 | 1235 |
| 35 | 56789 | 56789 | 1234 |
Highlighted rows have a single set. Five cells is the largest cage our game deals, on Expert.
6-, 7-, 8- and 9-cell cages: read the digits left out
Past five cells it is easier to list what a cage leaves out. The nine digits add up to 45, so a cage of six cells making 39 leaves out three digits making 6, which can only be 1, 2 and 3. Every row of the chart works this way: a cage of k cells making a total is the mirror image of a cage of 9 − k cells making 45 minus that total, and the program checks the mirror for every row.
| Total | Digits left out, one set each | In every set | In no set |
|---|---|---|---|
| 21 | 789 | 123456 | 789 |
| 22 | 689 | 123457 | 689 |
| 23 | 589 679 | 1234 | 9 |
| 24 | 489 579 678 | 123 | — |
| 25 | 389 479 569 578 | 12 | — |
| 26 | 289 379 469 478 568 | 1 | — |
| 27 | 189 279 369 378 459 468 567 | — | — |
| 28 | 179 269 278 359 368 458 467 | — | — |
| 29 | 169 178 259 268 349 358 367 457 | — | — |
| 30 | 159 168 249 258 267 348 357 456 | — | — |
| 31 | 149 158 167 239 248 257 347 356 | — | — |
| 32 | 139 148 157 238 247 256 346 | — | — |
| 33 | 129 138 147 156 237 246 345 | — | — |
| 34 | 128 137 146 236 245 | 9 | — |
| 35 | 127 136 145 235 | 89 | — |
| 36 | 126 135 234 | 789 | — |
| 37 | 125 134 | 6789 | 1 |
| 38 | 124 | 356789 | 124 |
| 39 | 123 | 456789 | 123 |
| Total | Digits left out, one set each | In every set | In no set |
|---|---|---|---|
| 28 | 89 | 1234567 | 89 |
| 29 | 79 | 1234568 | 79 |
| 30 | 69 78 | 12345 | — |
| 31 | 59 68 | 12347 | — |
| 32 | 49 58 67 | 123 | — |
| 33 | 39 48 57 | 126 | — |
| 34 | 29 38 47 56 | 1 | — |
| 35 | 19 28 37 46 | 5 | — |
| 36 | 18 27 36 45 | 9 | — |
| 37 | 17 26 35 | 489 | — |
| 38 | 16 25 34 | 789 | — |
| 39 | 15 24 | 36789 | — |
| 40 | 14 23 | 56789 | — |
| 41 | 13 | 2456789 | 13 |
| 42 | 12 | 3456789 | 12 |
| Cells | Total | Set | Left out |
|---|---|---|---|
| 8 | 36 | 12345678 | 9 |
| 8 | 37 | 12345679 | 8 |
| 8 | 38 | 12345689 | 7 |
| 8 | 39 | 12345789 | 6 |
| 8 | 40 | 12346789 | 5 |
| 8 | 41 | 12356789 | 4 |
| 8 | 42 | 12456789 | 3 |
| 8 | 43 | 13456789 | 2 |
| 8 | 44 | 23456789 | 1 |
| 9 | 45 | 123456789 | — |
For six and seven cells each set is shown as the digits it does not contain: at 39, “123” means the cage is 4, 5, 6, 7, 8 and 9. Eight- and nine-cell cages have one set at every total.
Our game never deals a cage this big, but printed puzzles do, and the rule of 45 makes them for you: once the cages lying wholly inside a row are added up, the row’s remaining cells must make 45 minus that sum, with no digit repeated. That is a cage in all but outline, and this chart applies to it.
How do you use killer sudoku combinations on a real board?
The chart gives you the sets. The board then takes them away, and three checks do almost all the work:
- Strike any set that needs a digit no cell of the cage can take. The digits that set alone supplied go with it.
- Find a digit every surviving set needs. If the cage’s cells that could hold it share a row, column or box, nothing else in that row, column or box can.
- Keep a digit in a cell only if a set can still be finished around it. A 2 in a cage of 11 needs a 9 beside it; with no 9 available, the 2 goes.
Our solver calls the first and third cage arithmetic and runs them right after singles; the second is the killer form of locked candidates. Here they are on two real Hard boards from our generator, at the moment the solver uses them.
A cage whose set is settled is a naked pair or naked triple drawn for you: 2, 3 and 5 belong to those three cells, and the rest of the row and box can forget them.
Neither move needed the cage to have a single set. That is the point of the in every set column: on the board above, the 22 still had two possible sets when the 9 left row 1.
How often does each cage size and total turn up?
We dealt 300 boards at each level from our game’s generator and counted every cage. Bigger levels bring bigger cages, and bigger cages bring more sets to sort through:
| Level | Cages per board | Share of cages by size (1 / 2 / 3 / 4 / 5 cells) | Average sets per cage of 2+ cells |
|---|---|---|---|
| Easy | 58.3 | 61.1% / 38.9% | 2.9 |
| Medium | 37.0 | 17.7% / 45.8% / 36.5% | 4.2 |
| Hard | 32.5 | 18.8% / 34.1% / 26.2% / 20.9% | 5.3 |
| Expert | 29.1 | 19.0% / 28.1% / 22.0% / 17.1% / 13.7% | 5.8 |
300 boards at each level, dealt by our generator from fixed seeds. “Sets” counts the chart’s sets for each cage’s size and total, before anything is placed.
Which totals you meet follows the chart, too. Our generator fills a grid first and draws cages over it, so a cage’s total is whatever its digits happen to add up to. Every pair of different digits is about equally likely in a two-cell cage, and a total turns up roughly as often as it has pairs:
| Two-cell total | Pairs | If every pair were equally likely | On our boards |
|---|---|---|---|
| 3 or 17 | 1 each | 2.8% each | 2.8% and 2.9% |
| 4 or 16 | 1 each | 2.8% each | 2.7% and 2.7% |
| 5 or 15 | 2 each | 5.6% each | 5.4% and 5.8% |
| 6 or 14 | 2 each | 5.6% each | 5.9% and 5.5% |
| 7 or 13 | 3 each | 8.3% each | 8.5% and 8.3% |
| 8 or 12 | 3 each | 8.3% each | 8.2% and 8.2% |
| 9, 10 or 11 | 4 each | 11.1% each | 10.8%, 10.9% and 11.3% |
All 17,666 two-cell cages on the 1,200 boards above. A total and its mirror (3 and 17, 4 and 16, and so on) have the same number of pairs.
A total with one set is, by the same arithmetic, the least likely total to be dealt. On our boards 11.1% of two-cell cages had one, against 1 in 9 if every pair were equally likely; three-cell cages 5.0% (4.8% by the count of sets), four-cell 3.6% (3.2%) and five-cell 2.8% (3.2%). Learn them, but expect most cages to need the other columns.
Stuck on a cage with too many combinations?
A middle total in a big cage can have a dozen sets, and staring at it rarely helps. Work it from the outside instead:
- Subtract what is placed. A three-cell 15 with a 9 already in it is a two-cell 6, which is 15 or 24.
- Strike sets, not just candidates. Each digit the cage’s row, column or box already holds kills every set that needs it there, and a set is gone as soon as one of its digits has no cell left.
- After each strike, look for a digit in every surviving set. It belongs in the cage, as the 9 in the 22 above did.
- Make a smaller cage. The rule of 45 turns the leftover cells of a row, column or box into a cage with a known total, and the chart works on it like any other.
The one-set totals for two to five cells, sixteen numbers in all, plus the mirror trick for anything bigger: subtract the total from 45 and read the digits left out. For the order to use the rest in, level by level, see our Killer Sudoku tips and strategies; to check a puzzle from paper has exactly one answer, enter it in our Killer Sudoku solver.
For practice that starts gently, the daily Killer Sudoku is Easy on Mondays and Tuesdays: cages of one or two cells, and singles plus the two-cell chart finish every board.
Killer Sudoku
Four levels from our own generator, cages from one cell up to five, every board with exactly one answer.
▶ Play Killer Sudoku