Killer Sudoku Strategies: The 45 Rule, Innies and Outies

If you're new to killer, read the rules on our killer sudoku page first. In short, it's normal sudoku plus cages: each dashed cage shows the total of its squares, and no digit repeats inside a cage. Keep our cage combinations table open too, because the strategies below lean on it.

Strategy 1: the rule of 45

The digits 1 to 9 add up to 45. Every row, every column and every 3×3 box holds each of those digits once, so every row, column and box adds up to 45. Two rows together add up to 90, three to 135, and the same goes for columns and boxes.

On its own that's just arithmetic. It becomes useful when cages almost line up with a row, column or box. Then the difference between 45 and the cage totals tells you the value of the squares that don't fit. Those squares are called innies and outies.

Strategy 2: innies

An innie is a square inside a box (or row or column) that belongs to a cage reaching outside it, when every other square of the box is covered by cages lying entirely inside.

Here is an example from the centre box of a real grid. Each letter is a cage. Cage E starts in this box and continues one square to the right, outside the box.

• Cage A (3 squares, top row of the box): total 10
• Cage B (2 squares, left column, lower two): total 13
• Cage C (3 squares: the centre, bottom middle and bottom right): total 14
• Cage E (2 squares: the middle-right square of the box and the square to its right, outside the box): total 10

Cages A, B and C lie entirely inside the box and cover eight of its nine squares. Together they add up to 10 + 13 + 14 = 37. The box must total 45, so the one square left, E's square inside the box, is 45 − 37 = 8.

And it doesn't stop there. Cage E totals 10 and its first square is 8, so the square outside the box is 2. One subtraction gave two digits.

Strategy 3: outies

An outie is the mirror image: cages cover the whole box and stick out by one square. The cage totals add up to more than 45, and the excess is the square outside.

Same centre box, different cages. This time cage D runs down the right-hand column of the box and one square further, into the box below.

• Cage A (top row of the box): 10
• Cage B (left column, lower two): 13
• Cage C (middle column, lower two): 8
• Cage D (right column, lower two, plus the square below the box): 17

Together the four cages total 10 + 13 + 8 + 17 = 48. Inside the box they cover all nine squares, which must total 45. The extra 3 has to come from the one square outside the box, so that square is 3.

Innies and outies on rows and columns

The same subtraction works for any unit, and for groups of units:

• A row where cages inside it cover all but one or two squares: the leftover squares add up to 45 minus those cages.
• Two rows (90) or three rows (135) together: useful when cages zigzag between neighbouring rows and never fit a single one.
• Two leftover squares don't give digits straight away, but they give a total. Two squares adding to 3 must be 1 and 2. Treat them like a small imaginary cage and look up the total in the combinations table.

A useful habit is to sweep the grid once at the start: every row, every column, every box. Note where cages fit neatly, and do the subtraction wherever one or two squares are left over.

Strategy 4: cages that limit each other

Cages in the same row, column or box share the same digits 1 to 9, so they constrain each other. A worked example:

A row contains a 2-square cage with total 3. That cage can only be 1 and 2. The same row also contains a 3-square cage with total 12. On its own, 12 in three squares has seven possible sets: 1 2 9, 1 3 8, 1 4 7, 1 5 6, 2 3 7, 2 4 6 and 3 4 5. But the row's 1 and 2 are already used by the first cage, so the 12 cage can't contain either. Only one set is left: 3, 4 and 5.

The same thinking works with "always has" digits. A 3-square cage of 8 is either 1 2 5 or 1 3 4, so it always contains a 1. If it sits entirely inside one box, no other square in that box can be 1. That's a lot of crossing off from a single cage.

Strategy 5: use normal sudoku once digits appear

Killer is still sudoku. As soon as some digits are placed, the usual techniques apply: naked and hidden singles, locked candidates and pairs. The best killer solvers switch back and forth: cage logic to get a foothold, sudoku logic to spread it, then cage logic again as cages fill up.

When a cage is partly filled, update its total. A 3-square cage of 15 with a 9 already placed has two squares left that add to 6: 1 and 5, or 2 and 4.

About the killer puzzles on this site

Our killer sudoku generator starts from a full random grid and groups neighbouring squares into small cages, mostly of two to four squares, with no digit repeated in a cage. Cages may cross box borders, which is where innies and outies come in. Then it checks how many solutions the cages allow. If there is more than one, it reveals a single square where the solutions differ and checks again, until exactly one solution remains. That's why some puzzles start with a few digits given and others with none.

Because the cages are small, many totals near the ends of the range have only one set of digits, so a good first pass is to find those. The game's hint button also helps: in killer it fills a square for you and points to the cage combinations table.

A starting routine

1. Pencil in every single-square cage and every cage total with only one set of digits.
2. Cross off the "never has" digits in every cage (see the combinations table).
3. Sweep each box, row and column for innies and outies.
4. Look for cages in the same unit that limit each other.
5. Switch to normal sudoku techniques as digits appear, and update partly filled cages' remaining totals.

Then play a killer sudoku and try the routine on a fresh grid.