Killer Sudoku Cage Combinations

Killer Sudoku cage combinations are the short table every Killer solver keeps in their head: the cage sums that can be made in exactly one way. Spot one of those sums and you know the digits before you have written a single candidate.

What the pattern is

A cage is a group of cells with a target sum, and no digit may repeat inside it. Most sums split several ways, but at the extremes there is only one option. Three different digits adding to 6 can only be 1, 2 and 3, because nothing else fits into so small a total.

When a cage has just one combination, those digits are locked into it and every other candidate leaves its cells. If all the cage cells also sit in one unit, the digits leave the rest of that unit too, exactly as a naked set would.

The forced sums

Cells	Sum	The only possible digits
2	3	12
2	4	13
2	16	79
2	17	89
3	6	123
3	7	124
3	23	689
3	24	789
4	10	1234
4	11	1235
4	29	5789
4	30	6789

Five-cell cages behave the same way at 15, 16, 34 and 35. The pattern is easy to remember: the lowest possible total for a cage of any size uses the smallest digits, the highest uses the largest, and the sums one step in from each end have exactly one combination as well.

When to look for it

On your first pass, before any candidate marking. These sums are read off the printed clues.
On the smallest and largest sums on the grid, which is where single combinations live.
On every two-cell cage, since two-cell sums are the most restrictive on the board.
Again later, once eliminations have knocked digits out of a cage and cut its options down to one.

How to apply it, step by step

Sweep the grid for cages of two, three and four cells.
Check each sum against the minimum and maximum for that cage size. The minimum is 1 plus 2 plus 3 and so on, the maximum uses the biggest digits.
If the sum matches a forced row above, write those digits into the cage cells as their only candidates.
Erase every other candidate from those cells.
If the cage lies wholly in one row, column or box, erase those digits from the rest of that unit as well.
Re-check neighbouring cages and units, because a locked cage often exposes a single straight away.

A worked example

Example

Box 3, with a three-cell cage sitting across its top row:

   G   G   G       G: sum 7 in 3 cells  ->  only 1 + 2 + 4
   .   .   .
   .   .   .

  1, 2 and 4 leave the other six cells of box 3.
  The cage also lies in one row, so 1, 2 and 4 leave
  the rest of that row too.

  A two-cell cage of 17 anywhere on the grid is {8,9},
  so 8 and 9 leave every cell that shares a unit with both.

Note what the combination does not tell you: it fixes the set of digits, never their order inside the cage.

Common mistakes

Assuming positions. Knowing a cage is 1, 2 and 4 says nothing about which cell holds which digit until another rule decides.
Allowing repeats. A two-cell cage of 4 is 1 and 3, not 2 and 2. Digits never repeat inside a cage.
Eliminating outside a unit. The unit-wide erase only works when every cell of the cage shares that row, column or box. An L-shaped cage often shares a box but not a row.
Stopping after the first pass. Cages with two or three combinations become forced later, as placements rule options out.

Where you meet it

Cage combinations are the opening move on easy Killer Sudoku and remain the backbone of every Killer solve above it. Once you are fluent with the forced sums, the next step is working with cages that keep several combinations and squeezing information out of what those combinations share.