How to Solve Killer Sudoku

A killer sudoku gives you no digits, only cages with sums. That looks harder than it is. Almost every killer can be solved with four tools: cage combinations, the rule of 45, innies and outies, and ordinary sudoku logic. This page explains each tool and the order in which to use them. If you do not know the rules yet, read What is killer sudoku? first.

1. Write down the cage combinations

Each cage has a fixed number of cells and a fixed sum, and no digit repeats inside a cage. So every cage allows only a short list of digit sets. Learn the lists with one option, because those cages are solved before you start:

Cages with two options are nearly as good. A two-cell 5 is 1+4 or 2+3; a two-cell 15 is 6+9 or 7+8. Write the possible digits in the cells as pencil marks. A cell in a two-cell 16 can only be 7 or 9, whatever else happens in the grid.

Combinations also tell you what a cage cannot hold. A two-cell cage with sum 10 cannot contain 5, because the other cell would also have to be 5. A three-cell cage with sum 8 cannot contain 6, 7, 8 or 9.

2. Use the rule of 45

The digits 1 to 9 add up to 45. So every row, every column and every 3×3 box adds up to 45. This is the tool that makes killer sudoku different from classic sudoku, and most of the first digits in a puzzle come from it.

Look at a box. Add up the sums of the cages that lie completely inside it. If those cages cover eight of the nine cells, the ninth cell is 45 minus that total. For example, if four cages inside a box have sums 11, 9, 14 and 7, they add up to 41, so the remaining cell is 4.

The same works for a row or a column, and for two boxes together (90) or three boxes together (135). Two cells left over give you a sum for a pair of cells, which is a cage in all but name. Three cells left over give you a sum for three cells.

3. Innies and outies

Cages often cross the border of a box or a row. The rule of 45 still works. You only have to decide which cells are in and which are out.

An innie is a cell inside the box that belongs to a cage mostly outside the box. Add up all the cages that lie completely inside the box, then add the sum of any cage that reaches into the box. If exactly one cell of that cage is inside the box, the innie, then you cannot use its full sum. Instead, take the box total 45, subtract the sums of the cages fully inside, and the difference is the value of the innie.

An outie is the opposite: a cell outside the box that belongs to a cage mostly inside the box. Add up all cages that have at least one cell in the box, including the outie's cage. That total is 45 plus the outie. So the outie is the total minus 45.

Example: a box contains cages with sums 12, 15 and 8, and a fourth cage with sum 14 that has three cells in the box and one cell just outside. The total of all four cages is 49. The outie is 49 − 45 = 4.

Check rows and columns as well as boxes. Two rows together add up to 90, and a cage that sticks out of two rows by one cell gives you that cell at once.

4. Combine cages with sudoku rules

Once a few digits are placed, the ordinary sudoku rules do most of the work. A digit placed in a row removes it from every pencil-mark list in that row, and that often reduces a cage to a single combination.

Two cells in the same row, column or box that both hold the same two candidates, for example 7 and 9 in a two-cell 16 cage, form a pair. No other cell in that row, column or box can be 7 or 9. Killer sudoku produces these pairs all the time, because small cages with one combination are exactly that.

A cage whose cells all lie in one row or box removes its digits from the rest of that row or box, even before you know which cell gets which digit. A two-cell 3 cage inside a box means no other cell of that box is 1 or 2.

5. A solving order that works

  1. Mark every cage with one or two combinations.
  2. Go through all nine boxes with the rule of 45 and note every innie and outie. Do the same for the rows and columns that are almost covered by whole cages.
  3. Place the digits found this way and update the pencil marks.
  4. Look for pairs and triples created by the small cages.
  5. Repeat. Each placed digit changes the combinations of its cage and of every cage in its row, column and box.

If you get stuck, go back to the rule of 45 with the digits you now have. A box in which five cells are known leaves a sum for the other four, and that sum often has a single combination.

6. Mistakes to avoid

Play Killer Sudoku free →