1. Start with sums that have one combination
Sums like 3, 4, 16 and 17 (two cells) work only one way. Those cages give certainty and are always your starting point.
Killer sudoku combinations
Which digits can form sum 13 in a three-cell cage? This page is the complete reference: for each cage size, every possible sum with all combinations of different digits. Keep it beside your puzzle — whether you're just starting out or working through an Expert grid.
How to use the table
2. Cross out combinations
Is there already a 5 in the cage's row or box? Then every combination containing a 5 is out. Often only one option remains.
3. Watch for forced digits
If a digit appears in every remaining combination, it is certainly in the cage — even if you don't yet know which cell. You can already rule that digit out elsewhere in the row or column.
The unique sums — your best friends
These sums have exactly one combination. Spot them in a puzzle and the digits are fixed immediately:
Cages of 2 cells
- 3 = 1+2
- 4 = 1+3
- 16 = 7+9
- 17 = 8+9
Cages of 3 cells
- 6 = 1+2+3
- 7 = 1+2+4
- 23 = 6+8+9
- 24 = 7+8+9
Cages of 4 cells
- 10 = 1+2+3+4
- 11 = 1+2+3+5
- 29 = 5+7+8+9
- 30 = 6+7+8+9
Cages of 5 cells
- 15 = 1+2+3+4+5
- 16 = 1+2+3+4+6
- 34 = 4+6+7+8+9
- 35 = 5+6+7+8+9
All combinations per cage size
Each row shows a sum with all possible combinations of different digits 1 to 9. Remember: no digit repeats within a cage.
Cages of 2 cells
| Sum | Possible 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 |
Cages of 3 cells
| Sum | Possible 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 |
Cages of 4 cells
| Sum | Possible 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 |
Cages of 5 cells
| Sum | Possible 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 |
New to killer sudoku? First read the rules and solving tips — or print a bundle via print killer sudoku and keep this table beside it.
Frequently asked questions about the combinations
Why does sum 5 in two cells appear twice?
Sum 5 can be 1+4 or 2+3. The table shows both; which one applies follows from what the row, column and box already rule out.
Why do the tables stop at five cells?
Cages on BreinPlezier are at most five cells, and larger cages are rare in other killers too. Besides, larger cages are easier to work out in reverse: which digits are missing (the rule of 45).
Do I have to memorise this table?
No. The unique sums stick by themselves as you play; the rest you simply look up. Many players keep the table nearby for the first weeks and rarely need it afterwards.
May I print or share this table?
Yes, please do. Print the page or share the link with fellow puzzlers — it stays freely available.