Killer Sudoku Cage Combinations
Look up every digit combination for a Killer Sudoku cage — and see which 43 size-and-sum pairs fix the digits outright, which is the opening of every puzzle.
Forty-three cages are forced by their sum alone
Cage digits are distinct, so a cage of n cells summing to s is a choice of n different digits from 1 to 9. Enumerating all 511 non-empty subsets gives 129 distinct size-and-sum pairs, and 43 of them admit exactly one combination. Those are the cages you can fill in before doing any reasoning at all.
| Cells | Sums it can reach | Forced sums | How many |
|---|---|---|---|
| 1 | 1–9 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | 9 |
| 2 | 3–17 | 3, 4, 16, 17 | 4 |
| 3 | 6–24 | 6, 7, 23, 24 | 4 |
| 4 | 10–30 | 10, 11, 29, 30 | 4 |
| 5 | 15–35 | 15, 16, 34, 35 | 4 |
| 6 | 21–39 | 21, 22, 38, 39 | 4 |
| 7 | 28–42 | 28, 29, 41, 42 | 4 |
| 8 | 36–44 | 36, 37, 38, 39, 40, 41, 42, 43, 44 | 9 |
| 9 | 45–45 | 45 | 1 |
Every size from two to seven has exactly four, and they are always the two smallest sums and the two largest. Nothing in the middle of a size's range is ever forced. So the standard advice to attack the extreme cages first is not a rule of thumb — it is the complete list of cages that can be attacked that way.
Except at eight cells, where the reason is different
Size eight breaks the pattern, and structurally rather than arithmetically. An eight-cell cage is defined by the one digit it leaves out, so its sum is 45 minus that digit. Nine possible omissions, nine possible sums, every one unique — and they cover the entire range 36–44 rather than clustering at the edges. It is the only size where being forced is not an edge effect.
| Sum | Digit left out | Cage |
|---|---|---|
| 36 | 9 | 1 2 3 4 5 6 7 8 |
| 37 | 8 | 1 2 3 4 5 6 7 9 |
| 38 | 7 | 1 2 3 4 5 6 8 9 |
| 39 | 6 | 1 2 3 4 5 7 8 9 |
| 40 | 5 | 1 2 3 4 6 7 8 9 |
| 41 | 4 | 1 2 3 5 6 7 8 9 |
| 42 | 3 | 1 2 4 5 6 7 8 9 |
| 43 | 2 | 1 3 4 5 6 7 8 9 |
| 44 | 1 | 2 3 4 5 6 7 8 9 |
Sizes one and eight both give nine forced sums, by dual arguments: pick a digit, or leave one out. And the nine-cell cage is the degenerate case at 45 — the same 45 that every row, column and box has to total, which is the whole basis of the 45 rule.
The least useful clue in the game
4 cells summing to 20 is the worst cage you can be handed: 12 different combinations satisfy it, more than any other size-and-sum pair on the board. A cage tells you least when its sum sits in the middle of what its size can reach — the exact complement of why the extreme cages tell you most.
The twelve landmark values that Killer solvers commit to memory are checked here against the enumeration rather than transcribed from a reference, and all twelve agree:
| Cage | Digits | Cage | Digits |
|---|---|---|---|
| 2 cells, 3 | 1 2 | 3 cells, 23 | 6 8 9 |
| 2 cells, 4 | 1 3 | 3 cells, 24 | 7 8 9 |
| 2 cells, 16 | 7 9 | 4 cells, 10 | 1 2 3 4 |
| 2 cells, 17 | 8 9 | 4 cells, 11 | 1 2 3 5 |
| 3 cells, 6 | 1 2 3 | 4 cells, 29 | 5 7 8 9 |
| 3 cells, 7 | 1 2 4 | 4 cells, 30 | 6 7 8 9 |
Worth noting against KenKen, which is also built from cages and gets the key rule the other way round: a KenKen cage may repeat a digit, because its only restriction is one of each per row and column. Same shape on the page, opposite arithmetic underneath — and it is why these combination tables are short enough to be worth memorising while KenKen's would not be.
How to use
- Enter how many cells the cage has and what it sums to.
- The tool lists every combination of distinct digits that fits.
- A single combination means the cage is forced — start there.
- Browse the table of all 43 forced cages to spot them on your grid.
- Remember every row, column and box totals 45.
Frequently asked questions
What is a Killer Sudoku cage?
A group of cells with a printed sum and no other clue. The digits inside must be distinct and must also obey the ordinary sudoku rules for their rows, columns and boxes. Since the digits are distinct, a cage of n cells summing to s is simply a choice of n different digits from 1 to 9.
Which cages are forced by their sum alone?
Forty-three size-and-sum pairs admit exactly one combination. Nine of them are single cells, one is a whole nine-cell line summing to 45, and the rest are the two lowest and two highest sums at each size — 3, 4, 16 and 17 for two cells, 6, 7, 23 and 24 for three, 10, 11, 29 and 30 for four, and so on.
Why do the forced cages sit at the extremes?
Because a sum near the bottom of what a size can reach can only be made from the smallest digits, and a sum near the top only from the largest. Sums in the middle have room to be assembled several ways. Every size from two to seven has exactly four forced sums, and none of them is ever in the middle of that size range.
Why does a cage of eight cells behave differently?
Because an eight-cell cage is defined by the one digit it leaves out, so its sum is 45 minus that digit. Nine possible omissions give nine possible sums, and every single one is unique — covering the whole range from 36 to 44 rather than clustering at the edges. It is the only size where being forced is not an edge effect.
What is the least useful cage clue?
Four cells summing to 20, which twelve different combinations satisfy. A cage tells you least when its sum sits in the middle of what its size can reach, which is exactly the complement of why the extreme cages tell you most.
What is the 45 rule?
The digits 1 to 9 sum to 45, so every row, every column and every box of a Killer Sudoku totals 45. That lets you work out a missing cell by subtracting the cages that lie wholly inside a line from 45, and it is why a nine-cell cage always reads 45.
Is this the same as KenKen?
No, and the difference is the rule that matters most. A Killer cage holds distinct digits, which is what makes the combination tables on this page finite and short. A KenKen cage may repeat a digit, since its only restriction is one of each per row and column. Same shape on the page, opposite arithmetic underneath.
Does this send anything anywhere?
No. Every combination is enumerated in your browser and nothing is uploaded.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.