How to solve killer sudoku, step by step

The techniques that solve most killer sudokus: cage combinations, the rule of 45, and using sudoku rules on the candidates, with small examples.

Killer sudoku keeps the rules of sudoku: every row, column and box holds each number once. It adds cages, the dashed outlines with a small total in the corner. The numbers in a cage add up to its total, and a number never repeats inside a cage. Most killer sudokus start with no numbers at all, so everything comes from the cages.

Learn the cage combinations

Some cages can only be filled one way. A 3 in two cells is always 1 and 2. A 17 in two cells is always 8 and 9. A 6 in three cells is always 1, 2 and 3. You don’t know the order yet, but you know which numbers go in.

A 3, a 17 and a 6 cage, with the only numbers each can hold.

The useful ones are the smallest and largest totals for each cage size:

Cells	Smallest totals	Largest totals
2	3 = 1 + 2, 4 = 1 + 3	16 = 7 + 9, 17 = 8 + 9
3	6 = 1 + 2 + 3, 7 = 1 + 2 + 4	23 = 6 + 8 + 9, 24 = 7 + 8 + 9
4	10 = 1 + 2 + 3 + 4, 11 = 1 + 2 + 3 + 5	29 = 5 + 7 + 8 + 9, 30 = 6 + 7 + 8 + 9

The killer sudoku combinations tool lists every combination for any cage size and total.

No repeats inside a cage

Because a cage never repeats a number, some totals lose options. A 4 in two cells can’t be 2 + 2, so it’s 1 and 3. A 16 in two cells can’t be 8 + 8, so it’s 7 and 9.

The rule of 45

Every row, column and box holds 1 to 9 once, so its numbers add up to 45. Add the cages that lie fully inside one box. If one cell of the box is left over, it holds 45 minus that sum.

Four cages inside the box add up to 9 + 8 + 13 + 9 = 39. The last cell's cage sticks out of the box, so that cell is 45 − 39 = 6.

It works for rows and columns too, and for two or three boxes together (90 or 135). The cells left over are called innies when they’re inside the area and outies when they stick out of it. Either way, they add up to the difference.

Use sudoku rules on the candidates

Pencil in the numbers each cage allows. Then use plain sudoku: a number that’s already in a row, column or box can’t go anywhere else in it. When a cage is down to one place for a number, write it in.

Putting it together

Fill the cages with only one combination, at least as candidates.
Use the rule of 45 on every box, row and column with a cell or two left over.
Remove candidates with sudoku rules.
Use each new number to narrow down the cages around it.

Big cages with many combinations come last. By then the cells around them usually leave only one way to fill them.