Killer Sudoku: rules, cage sums and tips

Updated 2026-10-30

Killer Sudoku is Sudoku with arithmetic: the grid is divided into dotted "cages", each with a target sum, and the digits inside a cage must add up to that number without repeating. Usually there are few or no givens, so the cage sums do the work.

Killer Sudoku rules

All the normal Sudoku rules still apply: fill the 9×9 grid so that every row, column and 3×3 box contains the digits 1–9 exactly once. On top of that:

• The grid is covered by cages, groups of cells outlined with a dotted line.
• Each cage shows a small number in its corner: the sum of the digits in that cage.
• A digit may not repeat within a cage, even if the cage crosses box boundaries.

In most Killer Sudoku puzzles every cell belongs to exactly one cage, so the cage sums together add up to 405, the total of the whole grid (nine rows of 45).

The 45 rule

The single most useful idea in Killer Sudoku is the 45 rule. The digits 1 to 9 add up to 45, so every row, every column and every box must total 45. Two rows together total 90, three rows 135, and so on.

This lets you find cells the cages do not tell you about directly. Suppose the cages lying entirely inside one box add up to 38, and only one cell of that box belongs to a cage that sticks out. That lone cell must be 45 − 38 = 7.

Innies and outies

That example is called an innie: a cell inside a unit (row, column, box or a group of them) that is not covered by cages fully contained in the unit. An outie is the reverse: a cell outside the unit that belongs to a cage poking in from outside.

Example of an outie: imagine the cages that cover all nine cells of a row also include one extra cell in the row below, and those cages sum to 52. The row itself contributes 45, so the extra cell is 52 − 45 = 7.

When there are two innies or two outies you get their sum rather than their values, which works exactly like a new two-cell cage. Combined with the combination table below, that is often enough.

Cage combinations worth memorising

Some cage sums can be made in only one way. Spotting them is the fastest route into a puzzle. These are the unique combinations for cages of two to five cells:

Notice the symmetry: a combination of n cells and its "complement" of 9 − n cells always add up to 45. That is why a five-cell 15 cage (1–5) mirrors a four-cell 30 cage (6–9).

Other sums are not unique but still narrow things down. A two-cell cage of 10 can be 1+9, 2+8, 3+7 or 4+6, but never 5+5, so a 10 cage in two cells can never contain a 5. Likewise a two-cell cage can never contain half its sum when the sum is even: a two-cell 8 cannot hold a 4, and a two-cell 12 cannot hold a 6.

Tips for solving Killer Sudoku

1. Start with unique cages. Mark their digits as candidates straight away. A 3 in two cells is 1 and 2; if one of those cells also sees a 1 elsewhere, you are done.
2. Apply the 45 rule everywhere. Check each box, row and column for innies and outies, then try pairs and triples of rows or columns.
3. Use cage overlap with boxes. If a 17 cage (8 and 9) sits inside one box, no other cell in that box can be 8 or 9.
4. Think about what a cage cannot contain. A three-cell 23 cage needs 6, 8 and 9, so it has no room for a 7.
5. Fall back on normal Sudoku logic. Singles, pairs and pointing work exactly as in classic Sudoku once you have some candidates.
6. Write down cage combinations in the margin for medium sums with only two or three options. Crossing them off one by one is surprisingly effective.

Killer Sudoku and its cousins

If you like the arithmetic side, Kakuro uses the same sum-and-no-repeat idea on a crossword-style grid. For something with no numbers at all but the same regional logic, try Star Battle: you place stars so that each row, column and region gets the right number, and stars never touch. Read the Star Battle rules or browse more puzzles like Sudoku.