Killer Sudoku Combinations: Every Cage Sum, 2 to 9 Cells

In Killer Sudoku, a cage’s digits never repeat, so each sum can only be made a limited number of ways. Knowing those ways is most of the game. This page lists every combination for every cage size, generated by checking all 511 sets of distinct digits from 1 to 9.

New to Killer? Start with Killer Sudoku Rules and How to Start, then come back here.

How to use the chart

Find the cage size, then the sum. The right-hand column lists every set of digits that fits. A combination tells you which digits are in the cage, never where: the order inside the cage is for the row, column and box rules to decide.

Bold entries are the cages with only one possible combination. Those are the first cages to look for in any Killer puzzle, because they tell you exactly which digits you’re dealing with.

The ones worth memorising

You don’t need the whole chart in your head. These come up constantly and cover most openings:

The pattern is easy to rebuild: the smallest and largest sums for any size have exactly one combination, and so does the sum one step in from each end.

Two more shortcuts:

• Big cages are easier to read by what’s missing. A 7-cell cage of 40 uses every digit except two that add to 5 (45 − 40), so it’s missing 1 and 4 or 2 and 3. An 8-cell cage is missing exactly one digit: 45 minus the sum.
• Common digits. In a 2-cell cage of 10, every combination is two different digits that sum to 10, so 5 is never in it. A 3-cell cage of 8 is 1 2 5 or 1 3 4, so it always contains a 1.

The full chart

2 cells

3 cells

4 cells

5 cells

6 cells

7 cells

8 cells

9 cells

Where the chart stops helping

The chart narrows each cage down to a set of digits. To place them, combine it with the ordinary sudoku rules and with the rule of 45: every row, column and box adds up to 45, so if cages cover all but one cell of a box, that cell is 45 minus the cages’ total. Killer Sudoku Rules and How to Start walks through it with examples.

One Sudoku’s Killer mode has three levels. Easy uses small cages of two or three cells and a few filled-in digits to start from. Medium and Hard print sums alone, with cages of up to four and five cells. Every board can be finished by logic, with no guessing.