Sixteen cage sums in Killer Sudoku can be filled in exactly one way. A 2-cell cage summing to 3 is always 1 and 2; a 4-cell cage summing to 30 is always 6, 7, 8 and 9. Spotting one of those hands you every digit in the cage before you have solved anything else.

The 16 Killer Sudoku cage sums worth memorising

Killer Sudoku looks like Sudoku with extra arithmetic, and beginners often treat the cage sums as a chore: add things up, check nothing repeats, move on. That is the slow way to play. The fast way is to notice that most sums are ambiguous and a few are not, and to learn the ones that are not.

Why a handful of sums decide the whole grid

A cage tells you two things: how many cells it covers, and what those cells add up to. Usually that leaves you plenty of options. A 3-cell cage summing to 15 can be filled eight different ways, which tells you almost nothing on its own.

But the arithmetic runs out at the edges. A 2-cell cage cannot sum to less than 1 + 2, and it cannot sum to more than 8 + 9. At those extremes there is only one set of digits that works — and the moment you recognise the sum, you know the digits without doing any Sudoku at all.

The sixteen one-answer sums

These are every cage of two to five cells whose sum can be made in exactly one way:

Cells	Sum	The only possible digits
2	3	1 2
2	4	1 3
2	16	7 9
2	17	8 9
3	6	1 2 3
3	7	1 2 4
3	23	6 8 9
3	24	7 8 9
4	10	1 2 3 4
4	11	1 2 3 5
4	29	5 7 8 9
4	30	6 7 8 9
5	15	1 2 3 4 5
5	16	1 2 3 4 6
5	34	4 6 7 8 9
5	35	5 6 7 8 9

The pattern is easier to hold in your head than the list. For a cage of any size, the lowest possible sum is the smallest digits packed together, and the second lowest is that same set with its top digit bumped by one. The two highest sums mirror it from the other end. That is four one-answer sums per cage size, every time.

Larger cages behave differently, and it is worth knowing why. An 8-cell cage is "all the digits except one", so *every* 8-cell sum has a single answer. A 9-cell cage is always 1 through 9 and always totals 45 — the fact the whole "rule of 45" technique is built on.

Learning these is not memorising trivia. It is learning where the puzzle gives digits away for free, so you spend your attention on the parts that actually need thinking.

What to do the moment you spot one

Write the digits into the cage as candidates — all of them, in every cell of the cage.
Strike those digits out of every other cell in the rows, columns and boxes the cage touches. This is usually where the real progress comes from, not inside the cage itself.
Check whether the cage sits inside a single row, column or box. If it does, the elimination is total: no other cell in that unit can hold any of those digits.
Only then go back to ordering the digits inside the cage.

Step 2 is the one people skip. A 2-cell cage summing to 17 is not interesting because it contains an 8 and a 9 — it is interesting because it removes 8 and 9 from up to fourteen other cells.

The sums that look useful and are not

Middling sums are where beginners lose time. Some to be wary of:

A 3-cell cage summing to 15 has eight possible sets.
A 4-cell cage summing to 20 has twelve.
A 5-cell cage summing to 25 has twelve.

None of those is worth enumerating by hand early in a puzzle. Come back to them once the surrounding cells have narrowed things down — that is when a twelve-way cage often collapses to two.

Where to look things up

You do not have to memorise anything beyond the sixteen above. For everything else there is a lookup table: our full cage combination chart lists every sum for every cage size from two to nine cells, and marks the one-answer rows. It is generated by the same program that builds this site, so there is nothing in it to mistype.

If Killer Sudoku itself is new to you, start with what Killer Sudoku is and how cages work. If you already know the rules and want the deduction techniques that come after the arithmetic — the ones that decide the *order* of digits inside a cage — those live in the technique guides, each with worked examples and practice puzzles checked by a solver.

And if you want a puzzle to try one of these on today, the daily challenge is free, needs no account, and publishes a full step-by-step solution the following day.