Killer Sudoku Cage Combinations: The Complete Table

Look up a cage by its number of cells and total. A two-cell cage totaling 17 must contain 8 and 9. That tells you which digits belong in the cage, but not which cell gets the 8.

The tables below cover standard 9×9 Killer Sudoku: digits 1–9, no repeated digit inside a cage. Ordinary row, column and 3×3 box restrictions still apply. A cage crossing a box boundary does not gain permission to repeat a digit.

Jump to the cage size you need: 1 cell, 2 cells, 3 cells, 4 cells, 5 cells, 6 cells, 7 cells, 8 cells, 9 cells.

Read a set without fixing its order

Braces such as {8,9} mean an unordered set. For the 17 cage, both arrangements below satisfy the arithmetic. Another clue is needed to choose between them.

When the table lists several sets, separate them before making notes. A two-cell total of 5 allows {1,4} or {2,3}. It does not mean that 1, 2, 3 and 4 all go into the cage. A row restriction might remove one of those sets, or leave a set while deciding its order.

These small cages are useful to recognize without a lookup:

Use cell restrictions after the sum

Consider a three-cell cage totaling 6. Its only digit set is {1,2,3}. Suppose the left and middle cells can each take only 1 or 2, while the right cell can take 1, 2 or 3.

There are only two arrangements consistent with those notes: 1,2,3 and 2,1,3. Both put 3 on the right, so that placement is safe. The first two cells remain unresolved.

The candidate restrictions in this example are stated as premises. On a real board, obtain them from the row, column and box clues; do not erase a note merely to make a cage fit a convenient pattern. The classic Sudoku rules explain those shared restrictions.

You can practice the same row-and-box reasoning in classic Sudoku. That linked game has no sum cages; the table here is a separate reference for Killer Sudoku.

If part of the cage is already filled

Subtract the filled digits from the clue and count the remaining empty cells. Then remove any listed set that would repeat a digit already used in that cage.

For a three-cell cage totaling 6 with a 1 filled in, the two remaining cells total 5. The two-cell table offers {1,4} and {2,3}. Discard {1,4}: another 1 would repeat the filled digit. The remaining cells contain 2 and 3, with their order still to be determined.

Try a similar deduction

A three-cell cage totals 23. Its first two cells each have candidates {6,8}; its third cell has {6,8,9}. What can you place, and what remains undecided?

See the answer

The three-cell table gives only {6,8,9}. The third cell must be 9, because neither of the first two cells can take it. The remaining pair can be 6,8 or 8,6. No part of this argument chooses between those two orders.

Complete table by cage size

Each row lists every distinct digit set for that cell count and sum, before other clues are applied. A semicolon separates alternative sets. Individual digits inside a set are written in ascending order, not in board order.

All 511 nonempty subsets of the digits 1–9 are included. Nine are single digits; 502 use two or more digits. This includes arithmetic possibilities, not a claim that every set can be placed in every particular puzzle.

1-cell cage combinations

2-cell cage combinations

3-cell cage combinations

4-cell cage combinations

5-cell cage combinations

6-cell cage combinations

7-cell cage combinations

8-cell cage combinations

9-cell cage combinations

When a total is missing

A missing total for the chosen cage size is impossible under these rules. Two cells cannot total 18: the largest distinct pair is 8+9=17. Using 9 twice would reach 18 but break the cage rule.

For a cage of n cells, the smallest sum comes from 1 through n; the largest comes from the n largest digits. Eight-cell cages have another shortcut: the omitted digit is 45 minus the cage total. An eight-cell total of 37 therefore contains every digit except 8. The nine-cell total is always 45.

These statements apply to cages whose digits cannot repeat. They should not be carried over unchanged to a variant with different digits or repeat rules.

Watch cage arithmetic in use

Mistercorzi's Killer Sudoku Tutorial for Beginners No1 – Rules, Cages & 8 Stop Solve Exercises introduces the cage rules and small sum combinations. The player starts at the two-cell arithmetic chapter, 3:53; three-cell arithmetic follows at 5:14. Its puzzle examples are separate from the local diagrams above.

Watch on YouTube

Rules and references

Conceptis's Killer Sudoku rules and Krazydad's instructions specify the sum and no-repeat conditions. SudokuWiki's combination reference discusses how digit sets constrain cages. The table and local examples here were calculated separately; none of those publishers' puzzle boards or illustrations were reused.