How to solve Killer Sudoku: combinations, cages and the rule of 45

8 min read · Updated October 13, 2026

Killer Sudoku keeps every rule of ordinary Sudoku and adds one more: the grid is divided into dashed cages, and the digits in each cage must add up to the small number in its corner. Many Killer puzzles start with an empty grid, so all the information is in those sums.

That sounds like arithmetic, but you'll only ever add small numbers. The real skill is knowing which digits can make a sum, and one simple fact about rows, columns and boxes that unlocks most hard puzzles: the rule of 45. Every diagram below is a real position from a Killer Sudoku on Puzzimo.

The rules

Fill every cell with a digit from 1 to 9.
Each row, column and 3×3 box contains every digit exactly once, as in classic Sudoku.
The digits in each cage add up to the cage's sum, and no digit repeats inside a cage.

If you're new to Sudoku itself, start with our guide to classic Sudoku: scanning, singles and pairs all work here too. This guide covers what's special about the cages.

How to solve Sudoku

Cage combinations

Because a cage can't repeat a digit, some sums can be made in only one way. Three cells adding up to 7 must be 1, 2 and 4: 1 + 2 + 3 is only 6, and anything else is too big. You don't know the order yet, but you know exactly which digits the cage holds.

The 7 cage in the centre box has three cells. The only three different digits that add up to 7 are 1, 2 and 4, so those are the only candidates in all three cells.

These one-way sums are worth learning by heart. They are always the smallest and largest sums for their cage size, plus the next one in:

2 cells: 3 = 1 + 2 · 4 = 1 + 3 · 16 = 7 + 9 · 17 = 8 + 9
3 cells: 6 = 1 + 2 + 3 · 7 = 1 + 2 + 4 · 23 = 6 + 8 + 9 · 24 = 7 + 8 + 9
4 cells: 10 = 1 + 2 + 3 + 4 · 11 = 1 + 2 + 3 + 5 · 29 = 5 + 7 + 8 + 9 · 30 = 6 + 7 + 8 + 9
5 cells: 15 = 1 + 2 + 3 + 4 + 5 · 16 = 1 + 2 + 3 + 4 + 6 · 34 = 4 + 6 + 7 + 8 + 9 · 35 = 5 + 6 + 7 + 8 + 9

Locked cages

When a one-way cage sits entirely inside one box, row or column, its digits are locked there. The rest of that box, row or column can't contain any of them. It's the Killer version of a naked pair, and you get it for free, before writing a single note.

The 3 cage at the bottom of the middle-right box can only be 1 + 2. Whichever way round they go, the box gets its 1 and its 2 there, so no other cell of the box can hold either digit.

The same works for any cage whose digits you know: a 17 cage inside a row takes that row's 8 and 9, and a 24 cage inside a box takes its 7, 8 and 9.

The rule of 45

Every row, column and box holds the digits 1 to 9 once, and 1 + 2 + … + 9 = 45. Cages often don't line up with the boxes, and that's exactly where this helps. Add up the cages that cover a box: if they cover one cell too many, the extra cell (an outie) is the total minus 45.

Three cages cover the bottom-left box: 18 + 11 + 22 = 51. The box itself adds up to 45, so the one cell of the 22 cage that sticks out of the box (row 9, column 4) must be 51 − 45 = 6.

It works the other way round too. If the cages lying wholly inside a box cover every cell but one, that last cell (an innie) is 45 minus their total.

Now look at the bottom-middle box. The 11, 20 and 8 cages lie entirely inside it and add up to 39. They cover every cell but one, so that cell is 45 − 39 = 6: the same answer, reached from the other side.

Using 45 on bigger regions

Nothing limits the rule to one unit. Two rows add up to 90, three columns to 135, and any group of whole boxes to 45 per box. When a long cage runs across two rows, add both rows together and the sums often leave just one or two cells to work out.

When two cells are left over, the rule gives their total instead of a single value. That's still useful: two cells that must add up to 3 are 1 and 2, and two that add up to 17 are 8 and 9, which brings you straight back to the combination table.

A solving routine

Look for one-way cages and write their digits as notes.
Lock any one-way cage that sits inside a single row, column or box.
Apply the rule of 45 to every row, column and box, looking for innies and outies.
Use ordinary Sudoku scanning on the digits you've placed.
Write full notes, removing candidates that can't make a cage's sum with its other cells.
Go back to step 3: every new digit can turn a cage into a one-way cage.

As in classic Sudoku, never guess. A proper Killer has exactly one solution, so there's always a deduction to find, and every Killer Sudoku on Puzzimo is checked for that.

Practise on the right level

Puzzimo's Killer Sudoku has four levels. Easy starts with at least 18 digits placed and small cages of two or three cells. Medium starts with at least 6 digits and cages of up to four. Hard (cages of up to five cells) and Expert (three to six) start with only the digits, if any, needed to make the solution unique, so the rule of 45 does most of the work.

Every puzzle is free. Notes, undo and hints are built in (a hint adds 30 seconds to a ranked time), a wrong digit counts as a mistake and the third one ends the round, and there's a new daily challenge every day.

Play Killer Sudoku now

Frequently asked questions

Can digits repeat in a Killer Sudoku cage?

No. A cage never contains the same digit twice, even when the cage crosses a box boundary. That's what makes the one-way sums work: three cells adding up to 6 must be 1, 2 and 3.

What is the rule of 45?

Each row, column and box contains 1 to 9 once, so it adds up to 45. If the cages covering a box cover one extra cell, that cell equals their total minus 45; if the cages inside a box miss one cell, that cell equals 45 minus their total.

Is Killer Sudoku harder than regular Sudoku?

It usually starts with fewer or no digits, so it feels harder at first. Once you know the one-way sums and the rule of 45, most of the work is quick addition and ordinary Sudoku logic.