Killer Sudoku Solver & Cage Combination Calculator
Enter a cage sum and cell count to see every valid combination of distinct digits 1–9 — the core of killer sudoku solving. Filter by digits you know are in (or out of) the cage, and use the full combination reference table below. Free, instant, and entirely in your browser.
Click a digit once to require it, twice to exclude it, a third time to clear.
A 3-cell cage can total 6 to 24.
No digit appears in every combination yet — add filters as you learn more from crossing rows and columns.
What Is a Killer Sudoku Cage Combination?
Killer sudoku adds arithmetic to classic sudoku: the grid is divided into dotted-outline cages, each printed with a target sum, and the cells of a cage must hold distinct digits that add up to that sum (on top of the usual row, column, and box rules). Solving killer sudoku is mostly about knowing which digit sets are even possible for each cage. A 2-cell cage totaling 3 can only be 1+2. A 3-cell cage totaling 15 has eight possibilities, but a 3-cell cage totaling 6 is forced to 1+2+3. This calculator lists every valid combination instantly, so you can spend your effort on the logic instead of the arithmetic.
The Rule of 45 (and Other Cage Math)
Rule of 45
Every row, column, and 3×3 box contains the digits 1–9, which always sum to 45 (and any two rows sum to 90, three to 135, and so on). If a set of cages exactly covers a box except for one cell, that cell’s value is 45 minus the cage totals (an innie). If the cages spill one cell outside the box, the extra cell is the cage totals minus 45 (an outie). Scanning for innies and outies is usually the first thing an experienced killer sudoku solver does.
Forced cages first
Cages at the extremes of their range have exactly one combination — for two cells: 3, 4, 16, and 17; for three cells: 6, 7, 23, and 24; for four cells: 10, 11, 29, and 30. Pencil those digits in immediately; they constrain everything they touch.
Shared digits
When every remaining combination for a cage contains the same digit, that digit is certain even if the full combination is not. The calculator points these out under the results. Combine this with the include/exclude filters as crossing rows and columns eliminate candidates.
No repeats in a cage
Cage digits are always distinct, even when a cage spans two boxes. That is why a 2-cell cage can never total 2, 4 can never be 2+2, and a 9-cell cage totaling 45 is always exactly the digits 1 through 9.
Killer Sudoku Combination Reference Table
Every possible cage combination for cage sizes 2 through 5. Combinations for 6-cell and larger cages are easiest read as their complement (a 6-cell cage in one box totals 45 minus the other three cells), or use the calculator above.
2-cell cages (sums 3–17)
| 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 (sums 6–24)
| 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 (sums 10–30)
| 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 (sums 15–35)
| 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 |
How to Use the Cage Calculator
- Type the cage sum — the small number in the corner of the cage.
- Pick the cage size — count the cells inside the dotted outline.
- Add filters as you learn — click a digit once if the cage must contain it, twice if a crossing row, column, or box rules it out.
- Read the results — one combination means a forced cage; when several remain, look for digits shared by all of them.
Frequently Asked Questions
Is the cage calculator free?
Yes, completely free with no signup. All the math runs in your browser, instantly.
What is the rule of 45?
Each row, column, and box sums to 45 because it contains the digits 1 through 9 exactly once. Comparing cage totals against 45 reveals the value of cells that stick into or out of a box — the famous innies and outies.
Can digits repeat inside a cage?
No. Killer sudoku cages always contain distinct digits, even when the cage crosses a box boundary. That constraint is what makes the combination lists on this page exhaustive.
Which cages should I start with?
Forced cages (one combination) and near-extreme sums, plus any innie or outie found with the rule of 45. Those give you certain digits that cascade through the rest of the grid.
Does my data leave my browser?
No. The calculator is plain JavaScript on your device. Nothing is uploaded, logged, or stored.
Can this solve a full killer sudoku?
It handles the cage math, which is the heart of killer solving; pair it with normal sudoku logic for the rest. For classic 9×9 puzzles, our step-by-step sudoku solver walks through the whole solution move by move.
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