Killer Sudoku Combinations

In Killer Sudoku, each dotted cage’s digits must add up to its clue with no repeats — so a cage’s size and sum fix which digit sets can fill it. This reference lists every combination, starting with the “no-choice” cages that crack a puzzle open.

Unique cages (one combination only)

When a cage’s sum can be made only one way, its digits are forced — pencil these in first.

Size	Sum	Digits
2 cells	3	1 2
2 cells	4	1 3
2 cells	16	7 9
2 cells	17	8 9
3 cells	6	1 2 3
3 cells	7	1 2 4
3 cells	23	6 8 9
3 cells	24	7 8 9
4 cells	10	1 2 3 4
4 cells	11	1 2 3 5
4 cells	29	5 7 8 9
4 cells	30	6 7 8 9
5 cells	15	1 2 3 4 5
5 cells	16	1 2 3 4 6
5 cells	34	4 6 7 8 9
5 cells	35	5 6 7 8 9
6 cells	21	1 2 3 4 5 6
6 cells	22	1 2 3 4 5 7
6 cells	38	3 5 6 7 8 9
6 cells	39	4 5 6 7 8 9
7 cells	28	1 2 3 4 5 6 7
7 cells	29	1 2 3 4 5 6 8
7 cells	41	2 4 5 6 7 8 9
7 cells	42	3 4 5 6 7 8 9
8 cells	36	1 2 3 4 5 6 7 8
8 cells	37	1 2 3 4 5 6 7 9
8 cells	38	1 2 3 4 5 6 8 9
8 cells	39	1 2 3 4 5 7 8 9
8 cells	40	1 2 3 4 6 7 8 9
8 cells	41	1 2 3 5 6 7 8 9
8 cells	42	1 2 4 5 6 7 8 9
8 cells	43	1 3 4 5 6 7 8 9
8 cells	44	2 3 4 5 6 7 8 9

The 45-rule

Every row, column and 3×3 box holds 1–9, so each always sums to 45. When cages fill a whole unit except one cell, that cell equals 45 minus the cage totals — and a cage poking a single cell into or out of a box (an innie or outie) is pinned the same way. It is the backbone of Killer solving; see the full Killer Sudoku guide.

Full combination tables

Every combination by cage size (2–5 cells, the common sizes). Each box is one legal digit set for that sum.

2-cell cages

Sum	Combinations
3	1 2
4	1 3
5	1 4 2 3
6	1 5 2 4
7	1 6 2 5 3 4
8	1 7 2 6 3 5
9	1 8 2 7 3 6 4 5
10	1 9 2 8 3 7 4 6
11	2 9 3 8 4 7 5 6
12	3 9 4 8 5 7
13	4 9 5 8 6 7
14	5 9 6 8
15	6 9 7 8
16	7 9
17	8 9

3-cell cages

Sum	Combinations
6	1 2 3
7	1 2 4
8	1 2 5 1 3 4
9	1 2 6 1 3 5 2 3 4
10	1 2 7 1 3 6 1 4 5 2 3 5
11	1 2 8 1 3 7 1 4 6 2 3 6 2 4 5
12	1 2 9 1 3 8 1 4 7 1 5 6 2 3 7 2 4 6 3 4 5
13	1 3 9 1 4 8 1 5 7 2 3 8 2 4 7 2 5 6 3 4 6
14	1 4 9 1 5 8 1 6 7 2 3 9 2 4 8 2 5 7 3 4 7 3 5 6
15	1 5 9 1 6 8 2 4 9 2 5 8 2 6 7 3 4 8 3 5 7 4 5 6
16	1 6 9 1 7 8 2 5 9 2 6 8 3 4 9 3 5 8 3 6 7 4 5 7
17	1 7 9 2 6 9 2 7 8 3 5 9 3 6 8 4 5 8 4 6 7
18	1 8 9 2 7 9 3 6 9 3 7 8 4 5 9 4 6 8 5 6 7
19	2 8 9 3 7 9 4 6 9 4 7 8 5 6 8
20	3 8 9 4 7 9 5 6 9 5 7 8
21	4 8 9 5 7 9 6 7 8
22	5 8 9 6 7 9
23	6 8 9
24	7 8 9

4-cell cages

Sum	Combinations
10	1 2 3 4
11	1 2 3 5
12	1 2 3 6 1 2 4 5
13	1 2 3 7 1 2 4 6 1 3 4 5
14	1 2 3 8 1 2 4 7 1 2 5 6 1 3 4 6 2 3 4 5
15	1 2 3 9 1 2 4 8 1 2 5 7 1 3 4 7 1 3 5 6 2 3 4 6
16	1 2 4 9 1 2 5 8 1 2 6 7 1 3 4 8 1 3 5 7 1 4 5 6 2 3 4 7 2 3 5 6
17	1 2 5 9 1 2 6 8 1 3 4 9 1 3 5 8 1 3 6 7 1 4 5 7 2 3 4 8 2 3 5 7 2 4 5 6
18	1 2 6 9 1 2 7 8 1 3 5 9 1 3 6 8 1 4 5 8 1 4 6 7 2 3 4 9 2 3 5 8 2 3 6 7 2 4 5 7 3 4 5 6
19	1 2 7 9 1 3 6 9 1 3 7 8 1 4 5 9 1 4 6 8 1 5 6 7 2 3 5 9 2 3 6 8 2 4 5 8 2 4 6 7 3 4 5 7
20	1 2 8 9 1 3 7 9 1 4 6 9 1 4 7 8 1 5 6 8 2 3 6 9 2 3 7 8 2 4 5 9 2 4 6 8 2 5 6 7 3 4 5 8 3 4 6 7
21	1 3 8 9 1 4 7 9 1 5 6 9 1 5 7 8 2 3 7 9 2 4 6 9 2 4 7 8 2 5 6 8 3 4 5 9 3 4 6 8 3 5 6 7
22	1 4 8 9 1 5 7 9 1 6 7 8 2 3 8 9 2 4 7 9 2 5 6 9 2 5 7 8 3 4 6 9 3 4 7 8 3 5 6 8 4 5 6 7
23	1 5 8 9 1 6 7 9 2 4 8 9 2 5 7 9 2 6 7 8 3 4 7 9 3 5 6 9 3 5 7 8 4 5 6 8
24	1 6 8 9 2 5 8 9 2 6 7 9 3 4 8 9 3 5 7 9 3 6 7 8 4 5 6 9 4 5 7 8
25	1 7 8 9 2 6 8 9 3 5 8 9 3 6 7 9 4 5 7 9 4 6 7 8
26	2 7 8 9 3 6 8 9 4 5 8 9 4 6 7 9 5 6 7 8
27	3 7 8 9 4 6 8 9 5 6 7 9
28	4 7 8 9 5 6 8 9
29	5 7 8 9
30	6 7 8 9

5-cell cages

Sum	Combinations
15	1 2 3 4 5
16	1 2 3 4 6
17	1 2 3 4 7 1 2 3 5 6
18	1 2 3 4 8 1 2 3 5 7 1 2 4 5 6
19	1 2 3 4 9 1 2 3 5 8 1 2 3 6 7 1 2 4 5 7 1 3 4 5 6
20	1 2 3 5 9 1 2 3 6 8 1 2 4 5 8 1 2 4 6 7 1 3 4 5 7 2 3 4 5 6
21	1 2 3 6 9 1 2 3 7 8 1 2 4 5 9 1 2 4 6 8 1 2 5 6 7 1 3 4 5 8 1 3 4 6 7 2 3 4 5 7
22	1 2 3 7 9 1 2 4 6 9 1 2 4 7 8 1 2 5 6 8 1 3 4 5 9 1 3 4 6 8 1 3 5 6 7 2 3 4 5 8 2 3 4 6 7
23	1 2 3 8 9 1 2 4 7 9 1 2 5 6 9 1 2 5 7 8 1 3 4 6 9 1 3 4 7 8 1 3 5 6 8 1 4 5 6 7 2 3 4 5 9 2 3 4 6 8 2 3 5 6 7
24	1 2 4 8 9 1 2 5 7 9 1 2 6 7 8 1 3 4 7 9 1 3 5 6 9 1 3 5 7 8 1 4 5 6 8 2 3 4 6 9 2 3 4 7 8 2 3 5 6 8 2 4 5 6 7
25	1 2 5 8 9 1 2 6 7 9 1 3 4 8 9 1 3 5 7 9 1 3 6 7 8 1 4 5 6 9 1 4 5 7 8 2 3 4 7 9 2 3 5 6 9 2 3 5 7 8 2 4 5 6 8 3 4 5 6 7
26	1 2 6 8 9 1 3 5 8 9 1 3 6 7 9 1 4 5 7 9 1 4 6 7 8 2 3 4 8 9 2 3 5 7 9 2 3 6 7 8 2 4 5 6 9 2 4 5 7 8 3 4 5 6 8
27	1 2 7 8 9 1 3 6 8 9 1 4 5 8 9 1 4 6 7 9 1 5 6 7 8 2 3 5 8 9 2 3 6 7 9 2 4 5 7 9 2 4 6 7 8 3 4 5 6 9 3 4 5 7 8
28	1 3 7 8 9 1 4 6 8 9 1 5 6 7 9 2 3 6 8 9 2 4 5 8 9 2 4 6 7 9 2 5 6 7 8 3 4 5 7 9 3 4 6 7 8
29	1 4 7 8 9 1 5 6 8 9 2 3 7 8 9 2 4 6 8 9 2 5 6 7 9 3 4 5 8 9 3 4 6 7 9 3 5 6 7 8
30	1 5 7 8 9 2 4 7 8 9 2 5 6 8 9 3 4 6 8 9 3 5 6 7 9 4 5 6 7 8
31	1 6 7 8 9 2 5 7 8 9 3 4 7 8 9 3 5 6 8 9 4 5 6 7 9
32	2 6 7 8 9 3 5 7 8 9 4 5 6 8 9
33	3 6 7 8 9 4 5 7 8 9
34	4 6 7 8 9
35	5 6 7 8 9

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Frequently asked questions

What are cage combinations in Killer Sudoku?

A cage combination is the set of digits that can fill a cage of a given size so they add up to its clue without repeating. For example a 2-cell cage summing to 5 is either {1,4} or {2,3}. Knowing the combinations a sum allows is the fastest way to pencil in candidates.

Which Killer cages have only one combination?

The extreme sums are forced. A 2-cell cage of 3 is {1,2} and of 17 is {8,9}; a 3-cell cage of 6 is {1,2,3} and of 24 is {7,8,9}; a 4-cell cage of 10 is {1,2,3,4} and of 30 is {6,7,8,9}. These "no-choice" cages are the best place to start a Killer puzzle.

What is the 45-rule in Killer Sudoku?

Every row, column and 3×3 box contains 1-9, so it always sums to 45. If cages cover a whole unit bar one cell, that cell is 45 minus the cage totals; a cage poking one cell into or out of a box (an innie or outie) is pinned the same way.