Killer Sudoku Number Combinations: Cage Sums and Locked Sets

A killer sudoku cage's number combinations are the sets of distinct digits 1–9 that add up to the cage's target sum. Since no digit may repeat within a cage, every sum has a fixed, checkable list of possible combinations — and some sums have only one. Knowing these sum combinations by heart is the arithmetic backbone of killer sudoku strategy. Below: the complete 2-cell and 3-cell tables, the locked single-combo sets, and the 45 rule.

What Cage Combinations Are

Killer sudoku keeps the classic 9×9 grid — every row, column, and 3×3 box must hold digits 1–9 (Sudoku) — and adds cages: groups of cells outlined together with a small target sum in the corner. Two rules govern every cage: the digits inside must add up to the target sum, and no digit may repeat within a single cage.

Because the digits are limited to 1–9 with no repeats, each sum has a mathematically fixed set of possible sum combinations. Learning the short lists — especially the sums with only one possible combination — is the fastest way to get better at killer sudoku.

The 45 Rule

Each row, column, and box contains the digits 1–9 exactly once, so every complete row, column, or box sums to 1+2+…+9 = 45. This gives killer sudoku its most powerful technique: compare a row's actual cage totals against 45, and the difference reveals what's hiding in the cells that spill across the boundary. It is simple addition — no advanced maths required. In practice, the 45 rule turns the cage-sum tables above from a reference into a weapon: once you know which combinations are possible, subtracting known cage totals from 45 tells you exactly what the leftover cells must contain.

2-Cell Cage Combinations (Complete)

Two-cell cages are the backbone of killer sudoku. There are only 36 possible pairs of distinct digits 1–9, spread across sums 3 to 17:

Notice the symmetry: sums pair up around 10 — sum 3 and sum 17 each have one combo, sum 4 and 16 each have one, and so on. That's because replacing each digit d with 10−d maps every combination of sum S to a combination of sum 20−S.

3-Cell Cage Combinations (Complete)

Three-cell cages cover sums 6 to 24, with 84 possible triples in total:

Locked Sets: Sums With Only One Combination

These are the combinations worth memorizing, because a locked cage narrows itself instantly — and once row, column, and box constraints place the digits, the cage fills itself:

How to Use Combinations While Solving

1. Pencil in locked cages first. A 2-cell cage summing to 16 can only be 7 and 9 — mark both cells as {7,9} immediately.
2. Combine cages with the 45 rule. If a row's cages sum to 42, the cells spilling into the neighbouring row must total 3 — and a 2-cell cage of sum 3 is locked to {1,2}.
3. Eliminate by overlap. A 2-cell cage summing to 9 ({1+8, 2+7, 3+6, 4+5}) sitting in a row that already shows an 8 can only be 2+7, 3+6, or 4+5.
4. Spot shared-digit cages. Two 2-cell cages in one unit summing to 16 ({7,9}) and 17 ({8,9}) must share the 9 — so exactly one of the four cells holds 9, and the other unit cells in those positions can't repeat the placed digits.

For more number-puzzle practice, see our sudoku puzzle book guide and the crossword books for adults roundup.

A Worked Example: Cracking a Cage With the 45 Rule

Suppose a row's cages total 42 across their first seven cells, and the row's last two cells form a single 2-cell cage. The row must sum to 45, so that pair has to total 45 − 42 = 3 — and a 2-cell cage summing to 3 is locked to {1, 2}. The cage is cracked without a single guess.

Now combine it with unit constraints: scan the column holding one of those cells. If the column already contains a 2, that cell must be 1 — and its partner is 9-free 2. Two locked facts, zero guessing. This is how combinations and the 45 rule work together in real solves.

Common Mistakes When Using Cage Combinations

Memorising the combination tables is only half the battle — the other half is avoiding the errors that trip up even experienced killer sudoku solvers. These are the ones to watch for.

Forgetting that combinations change with position. A 2-cell cage summing to 8 has the combinations (1,7), (2,6), (3,5) — but if that cage sits inside a row that already contains a 7, the (1,7) pair is dead. The tables list all possible combinations; your job is to cross off the ones the surrounding grid has already ruled out. Solvers who treat the table as the final answer instead of a starting filter waste the technique's power.

Misapplying the 45 rule across incomplete units. The 45 rule only works on a complete row, column, or box — or on a set of cages that exactly covers one. A frequent mistake is comparing cage totals against 45 for a row where one cage spills over into the next row. If the cages do not align exactly with the unit boundary, the arithmetic tells you nothing.

Ignoring the "innies and outies". When cages nearly — but not quite — cover a row, the difference between the cage totals and 45 tells you the value of the cell sticking out (an "outie") or the cell left uncovered (an "innie"). Many solvers learn the 45 rule but never learn this extension, which is where the technique earns its keep on harder puzzles.

Locking in a combination too early. A sum with a single combination is certain — but a sum with two or three is not. Writing a guess in ink because "it is probably (3,5)" is how clean grids die. Pencil marks exist precisely for this.