Killer Sudoku combinations: controlling the cages’ candidates

Cages and their sums are the defining features of Killer Sudoku. They are what make this variant of the classic number puzzle unique and a central part of the challenge and the solution of each grid. Cage combinations are thus nothing more than a step to solve these puzzles and one within the reach of every player. The lists below of Killer Sudoku combinations are thus not a cheat code or a strategy per se, but rather a memory aid to help players become more efficient and quicker when solving Killer challenges.

What are the Killer Sudoku cage combinations?

In the context of Killer Sudoku, cage combinations are the possible sets of numbers from 1 to 9 that players can use to reach a certain sum. In other words, they represent the candidates for cells and/or cages. These are not considered strategies or cheats, but simply a part of the process of solving the puzzles.

For example, for a 2-cell cage with a total sum of 5, there are only two possible combinations: 1,4 (1+4=5) or 2,3 (2+3=5). Therefore, 1,2,3, and 4 are the only feasible candidates for those two cells. Later, the players can then use this information to restrict the number of candidates to these even further. For instance, if number 1 is impossible in the cage of this example, then number 4 would also have to be excluded as a candidate for those cells.

List of Killer Sudoku combinations

The more cells a cage has and the higher its sum, the more complex and time-consuming it becomes to find and write down all the possible combinations. To help players become more efficient and quickly progress in their efforts to solve the grids, we have compiled a thorough list of all Killer Sudoku combinations.

2-cell cages    

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    

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    

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    

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    

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    

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    

36    12345678    
37    12345679    
38    12345689    
39    12345789    
40    12346789    
41    12356789    
42    12456789    
43    13456789    
44    23456789    

9-cell cages    

45    123456789

Unique cage combinations

Unique combinations are the most sought-after by players because they provide a partial solution to their cages and the overall grid. In simple terms, these refer to cases or cages where there is only one (thus the “unique”) possible combination.
For example, a 2-cell cage with a sum of 3 can only contain the naked pair 1,2. There is no other way to combine the numbers from 1 to 9, without repeating digits, to arrive at the value of 3. Another example would be a 4-cell cage with a sum of 30. It is only possible to arrive at that value by combining the numbers 6, 7, 8, and 9, which also makes it a unique combination.

Killer Sudoku combinations are a natural part of the process of solving these number puzzles. Remembering them all by heart is not easy, especially in the case of large cages with high values, which is why this complete list can be very useful for reducing your solving time. Besides, these cage combinations can go beyond cages themselves. They might also prove useful when players apply the 45-rule in Sudoku to reduce candidates rather than find the solution for a cell.

We hope this list can support players on their journey to becoming a master of Killer Sudoku puzzles.