Killer Combinations
Every cage in Killer Sudoku shows a target sum, and the first question is always the same: which digits can add up to that? Killer combinations — sometimes just called cage combinations — are the answer. Knowing them, or having them to hand, is the single biggest speed-up in Killer solving. This page is your reference table.
How to use the tables
Find your cage's size (number of cells) and its target sum. The table lists every set of digits that can fill it — remembering that digits inside a cage never repeat. If only one combination is listed, you've learned the cage's exact digits outright. If several are listed, you've still narrowed things enormously, and other techniques will pick the winner.
Two-cell cages
| Sum | Possible digits |
|---|---|
| 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} |
Three-cell cages
| Sum | Possible digits |
|---|---|
| 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} |
| … | (sums 11–19 have several each — see below for the extremes) |
| 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} |
The "one and only" combinations
These sums have exactly one combination, so a cage of that size and sum tells you its digits with certainty. They're the ones worth memorising:
| Cells | Sum → digits |
|---|---|
| 2 | 3 → {1,2} · 4 → {1,3} · 16 → {7,9} · 17 → {8,9} |
| 3 | 6 → {1,2,3} · 7 → {1,2,4} · 23 → {6,8,9} · 24 → {7,8,9} |
| 4 | 10 → {1,2,3,4} · 11 → {1,2,3,5} · 29 → {5,7,8,9} · 30 → {6,7,8,9} |
| 5 | 15 → {1,2,3,4,5} · 16 → {1,2,3,4,6} · 34 → {4,6,7,8,9} · 35 → {5,6,7,8,9} |
Putting them to work
Combinations rarely solve a cage alone — they hand you a short list, and the grid narrows it. A two-cell cage summing to 16 is {7, 9} for certain. A three-cell 8 is either {1,2,5} or {1,3,4} — and if a 4 already sits in the row, it must be {1,2,5}. Pair this reference with the rule of 45 and innies and outies and most Killer grids open up.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.