Killer Sudoku Help
Killer sudoku adds killer cages to sudoku grid. Each cage has a number which tells us sum of the digits in the cage. Digits within a cage cannot repeat.
In the image above we can conclude that digit outside the box in cage with sum 22 is 1 because we can sum all digits in the three cages and they add to 46, and sum of all cells in a box must be 45 (1+2+3+4+5+6+7+8+9).
Killer Cage Sudoku Solving Strategies
Innies and Outies
Main strategy for solving killer cages is with innes and outies. Innes are cells that are missing from a house (box, row, column) when all other cells are in cages that are fully in the house. When all cages in a house are in the house with exception of one or two cells, those cells are called outies. It sounds complicated but it's really not.
On image below we can see one innie cell marked with yellow, and one outie cell marked with blue.
Sum of all digits from 1 to 9 is 45. Sum of cages in box 1 is 40, hence yellow cell must be 5. Sum of cages in box 3 (including the outie cell) is 48, hence blue cell must be 3.
Yellow cell on image above must be 1, because sum of all cages is 136, and sum of 3 sets of digits 1 to 9 is 135 (first three rows or boxes).
Another way to find out that this is 1, is by summing all cages in box 1 including the outie cell. Sum is 46, outie (blue cell) must be 1.
Then we can sum all the cages in box 2, without orange outie cell, sum is 36, which means outie cell is 9.
Now if we sum all cells in box 3 including the the outie cells we get 55. Outies must sum to 10, and know orange cell is 9, which means yellow cell must be 1.
Combinations
Some cages sums have a single combination, for example 3 in two cell cage, 4 in two cell cage, 23 in three cell cage, 24 in three cells cage. Some have only two like 5 or 6 in two cell cage. See example below.
On the grid above, in box 1 killer cage with sum 3 must be 1 and 2, so we can remove 1 and 2 candidates from other cells in the same box and row. Same can be done on grid above for killer cages with sum 23, 24 and 4. In box 2 we have two cages that sum to 5, since we have only two combinations that sum to five, same candidates can be removed from other cells in the box.
Sometimes we can combine cages (when they are in the same house). On the grid above cages 5 and 6 sum to 11. Only combination that sums to 11 in four cells is 1235. Since 3 cannot go in cage 6 (because it will need another 3 to sum to 6) we can put 3 in cage with sum 5. Similarly for digit 5, it cannot go in cage that sums to 5 (other cell would have to be 0), we can put 5 in cage with sum 6. So the cage with sum 6 must be 1 and 5, and the cage with sum 5 must be 2 and 3.
Limiting one cell in the cage is often enough to find out combination of that cage. On the grid above cell R1C1 cannot be 15, which means the cage must be 24.
Below are combination tables separated by cage sizes:
Cage Size: 2
| Sum | Combinations |
|---|---|
| 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 |
Cage Size: 3
| Sum | Combinations |
|---|---|
| 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 |
Cage Size: 4
| Sum | Combinations |
|---|---|
| 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 |
Cage Size: 5
| Sum | Combinations |
|---|---|
| 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 |
Cage Size: 6
| Sum | Combinations |
|---|---|
| 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 |
Cage Size: 7
| Sum | Combinations |
|---|---|
| 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 |
Cage Size: 8
| Sum | Combination |
|---|---|
| 36 | 12345678 |
| 37 | 12345679 |
| 38 | 12345689 |
| 39 | 12345789 |
| 40 | 12346789 |
| 41 | 12356789 |
| 42 | 12456789 |
| 43 | 13456789 |
| 44 | 23456789 |
Cage Size: 9
| Sum | Combination |
|---|---|
| 45 | 123456789 |