Killer Sudoku cage combinations

Every cage sum can be built from a limited set of digit combinations. Memorise the highlighted ones (only one possibility) and use the rest to eliminate candidates.

All combinations use the digits 1–9 without repeats. Entries with a highlighted outline have exactly one possible combination.

Unique combinations to memorise

2 cells, sum 3: 1, 2
2 cells, sum 4: 1, 3
2 cells, sum 16: 7, 9
2 cells, sum 17: 8, 9
3 cells, sum 6: 1, 2, 3
3 cells, sum 7: 1, 2, 4
3 cells, sum 23: 6, 8, 9
3 cells, sum 24: 7, 8, 9
4 cells, sum 10: 1, 2, 3, 4
4 cells, sum 11: 1, 2, 3, 5
4 cells, sum 29: 5, 7, 8, 9
4 cells, sum 30: 6, 7, 8, 9
5 cells, sum 15: 1, 2, 3, 4, 5
5 cells, sum 16: 1, 2, 3, 4, 6
5 cells, sum 34: 4, 6, 7, 8, 9
5 cells, sum 35: 5, 6, 7, 8, 9
6 cells, sum 21: 1, 2, 3, 4, 5, 6
6 cells, sum 22: 1, 2, 3, 4, 5, 7
6 cells, sum 38: 3, 5, 6, 7, 8, 9
6 cells, sum 39: 4, 5, 6, 7, 8, 9

Try it now. CageDoku is a free, ad-free Killer Sudoku app with hints that explain the technique. Play in your browser.

2-cell cages

Sum	Combinations	#
3	1 2	1
4	1 3	1
5	1 42 3	2
6	1 52 4	2
7	1 62 53 4	3
8	1 72 63 5	3
9	1 82 73 64 5	4
10	1 92 83 74 6	4
11	2 93 84 75 6	4
12	3 94 85 7	3
13	4 95 86 7	3
14	5 96 8	2
15	6 97 8	2
16	7 9	1
17	8 9	1

3-cell cages

Sum	Combinations	#
6	1 2 3	1
7	1 2 4	1
8	1 2 51 3 4	2
9	1 2 61 3 52 3 4	3
10	1 2 71 3 61 4 52 3 5	4
11	1 2 81 3 71 4 62 3 62 4 5	5
12	1 2 91 3 81 4 71 5 62 3 72 4 63 4 5	7
13	1 3 91 4 81 5 72 3 82 4 72 5 63 4 6	7
14	1 4 91 5 81 6 72 3 92 4 82 5 73 4 73 5 6	8
15	1 5 91 6 82 4 92 5 82 6 73 4 83 5 74 5 6	8
16	1 6 91 7 82 5 92 6 83 4 93 5 83 6 74 5 7	8
17	1 7 92 6 92 7 83 5 93 6 84 5 84 6 7	7
18	1 8 92 7 93 6 93 7 84 5 94 6 85 6 7	7
19	2 8 93 7 94 6 94 7 85 6 8	5
20	3 8 94 7 95 6 95 7 8	4
21	4 8 95 7 96 7 8	3
22	5 8 96 7 9	2
23	6 8 9	1
24	7 8 9	1

4-cell cages

Sum	Combinations	#
10	1 2 3 4	1
11	1 2 3 5	1
12	1 2 3 61 2 4 5	2
13	1 2 3 71 2 4 61 3 4 5	3
14	1 2 3 81 2 4 71 2 5 61 3 4 62 3 4 5	5
15	1 2 3 91 2 4 81 2 5 71 3 4 71 3 5 62 3 4 6	6
16	1 2 4 91 2 5 81 2 6 71 3 4 81 3 5 71 4 5 62 3 4 72 3 5 6	8
17	1 2 5 91 2 6 81 3 4 91 3 5 81 3 6 71 4 5 72 3 4 82 3 5 72 4 5 6	9
18	1 2 6 91 2 7 81 3 5 91 3 6 81 4 5 81 4 6 72 3 4 92 3 5 82 3 6 72 4 5 73 4 5 6	11
19	1 2 7 91 3 6 91 3 7 81 4 5 91 4 6 81 5 6 72 3 5 92 3 6 82 4 5 82 4 6 73 4 5 7	11
20	1 2 8 91 3 7 91 4 6 91 4 7 81 5 6 82 3 6 92 3 7 82 4 5 92 4 6 82 5 6 73 4 5 83 4 6 7	12
21	1 3 8 91 4 7 91 5 6 91 5 7 82 3 7 92 4 6 92 4 7 82 5 6 83 4 5 93 4 6 83 5 6 7	11
22	1 4 8 91 5 7 91 6 7 82 3 8 92 4 7 92 5 6 92 5 7 83 4 6 93 4 7 83 5 6 84 5 6 7	11
23	1 5 8 91 6 7 92 4 8 92 5 7 92 6 7 83 4 7 93 5 6 93 5 7 84 5 6 8	9
24	1 6 8 92 5 8 92 6 7 93 4 8 93 5 7 93 6 7 84 5 6 94 5 7 8	8
25	1 7 8 92 6 8 93 5 8 93 6 7 94 5 7 94 6 7 8	6
26	2 7 8 93 6 8 94 5 8 94 6 7 95 6 7 8	5
27	3 7 8 94 6 8 95 6 7 9	3
28	4 7 8 95 6 8 9	2
29	5 7 8 9	1
30	6 7 8 9	1

5-cell cages

Sum	Combinations	#
15	1 2 3 4 5	1
16	1 2 3 4 6	1
17	1 2 3 4 71 2 3 5 6	2
18	1 2 3 4 81 2 3 5 71 2 4 5 6	3
19	1 2 3 4 91 2 3 5 81 2 3 6 71 2 4 5 71 3 4 5 6	5
20	1 2 3 5 91 2 3 6 81 2 4 5 81 2 4 6 71 3 4 5 72 3 4 5 6	6
21	1 2 3 6 91 2 3 7 81 2 4 5 91 2 4 6 81 2 5 6 71 3 4 5 81 3 4 6 72 3 4 5 7	8
22	1 2 3 7 91 2 4 6 91 2 4 7 81 2 5 6 81 3 4 5 91 3 4 6 81 3 5 6 72 3 4 5 82 3 4 6 7	9
23	1 2 3 8 91 2 4 7 91 2 5 6 91 2 5 7 81 3 4 6 91 3 4 7 81 3 5 6 81 4 5 6 72 3 4 5 92 3 4 6 82 3 5 6 7	11
24	1 2 4 8 91 2 5 7 91 2 6 7 81 3 4 7 91 3 5 6 91 3 5 7 81 4 5 6 82 3 4 6 92 3 4 7 82 3 5 6 82 4 5 6 7	11
25	1 2 5 8 91 2 6 7 91 3 4 8 91 3 5 7 91 3 6 7 81 4 5 6 91 4 5 7 82 3 4 7 92 3 5 6 92 3 5 7 82 4 5 6 83 4 5 6 7	12
26	1 2 6 8 91 3 5 8 91 3 6 7 91 4 5 7 91 4 6 7 82 3 4 8 92 3 5 7 92 3 6 7 82 4 5 6 92 4 5 7 83 4 5 6 8	11
27	1 2 7 8 91 3 6 8 91 4 5 8 91 4 6 7 91 5 6 7 82 3 5 8 92 3 6 7 92 4 5 7 92 4 6 7 83 4 5 6 93 4 5 7 8	11
28	1 3 7 8 91 4 6 8 91 5 6 7 92 3 6 8 92 4 5 8 92 4 6 7 92 5 6 7 83 4 5 7 93 4 6 7 8	9
29	1 4 7 8 91 5 6 8 92 3 7 8 92 4 6 8 92 5 6 7 93 4 5 8 93 4 6 7 93 5 6 7 8	8
30	1 5 7 8 92 4 7 8 92 5 6 8 93 4 6 8 93 5 6 7 94 5 6 7 8	6
31	1 6 7 8 92 5 7 8 93 4 7 8 93 5 6 8 94 5 6 7 9	5
32	2 6 7 8 93 5 7 8 94 5 6 8 9	3
33	3 6 7 8 94 5 7 8 9	2
34	4 6 7 8 9	1
35	5 6 7 8 9	1

6-cell cages

Sum	Combinations	#
21	1 2 3 4 5 6	1
22	1 2 3 4 5 7	1
23	1 2 3 4 5 81 2 3 4 6 7	2
24	1 2 3 4 5 91 2 3 4 6 81 2 3 5 6 7	3
25	1 2 3 4 6 91 2 3 4 7 81 2 3 5 6 81 2 4 5 6 7	4
26	1 2 3 4 7 91 2 3 5 6 91 2 3 5 7 81 2 4 5 6 81 3 4 5 6 7	5
27	1 2 3 4 8 91 2 3 5 7 91 2 3 6 7 81 2 4 5 6 91 2 4 5 7 81 3 4 5 6 82 3 4 5 6 7	7
28	1 2 3 5 8 91 2 3 6 7 91 2 4 5 7 91 2 4 6 7 81 3 4 5 6 91 3 4 5 7 82 3 4 5 6 8	7
29	1 2 3 6 8 91 2 4 5 8 91 2 4 6 7 91 2 5 6 7 81 3 4 5 7 91 3 4 6 7 82 3 4 5 6 92 3 4 5 7 8	8
30	1 2 3 7 8 91 2 4 6 8 91 2 5 6 7 91 3 4 5 8 91 3 4 6 7 91 3 5 6 7 82 3 4 5 7 92 3 4 6 7 8	8
31	1 2 4 7 8 91 2 5 6 8 91 3 4 6 8 91 3 5 6 7 91 4 5 6 7 82 3 4 5 8 92 3 4 6 7 92 3 5 6 7 8	8
32	1 2 5 7 8 91 3 4 7 8 91 3 5 6 8 91 4 5 6 7 92 3 4 6 8 92 3 5 6 7 92 4 5 6 7 8	7
33	1 2 6 7 8 91 3 5 7 8 91 4 5 6 8 92 3 4 7 8 92 3 5 6 8 92 4 5 6 7 93 4 5 6 7 8	7
34	1 3 6 7 8 91 4 5 7 8 92 3 5 7 8 92 4 5 6 8 93 4 5 6 7 9	5
35	1 4 6 7 8 92 3 6 7 8 92 4 5 7 8 93 4 5 6 8 9	4
36	1 5 6 7 8 92 4 6 7 8 93 4 5 7 8 9	3
37	2 5 6 7 8 93 4 6 7 8 9	2
38	3 5 6 7 8 9	1
39	4 5 6 7 8 9	1

How to use the table

Find the cage size and sum, list the combinations, then remove candidates that appear in none of them. If the cage is also restricted by its row or box (for example one cell must be 8), cross out combinations that do not contain that digit. Practice this in CageDoku — its hints name the exact cage that forces a digit.

Larger cages (7–9 cells) are easier to think of as the complement: a 7-cell cage with sum 40 leaves out two digits that add to 5 (1+4 or 2+3).