Killer strategy: cage sums and the rule of 45

Play Killer

Killer mixes two kinds of thinking: arithmetic inside each cage, and Sudoku logic across rows, columns and boxes. The rule of 45 is where the two meet, and it's the move that opens almost every Killer.

How the difficulty works

Each tier is set by the hardest step you'll need:

Easy needs only cage sums, singles and the rule of 45 for one cell at a time.
Medium needs the rule of 45 for two or three cells at once, a cage that claims a digit for a row, column or box, or Sudoku's pointing and pairs.
Hard needs the rule of 45 across several units at once (two or three rows or columns, or two boxes), or Sudoku's triples and X-Wing.

Nothing ever needs a guess.

Cages that can only be made one way

Most sums can be made several ways, but the smallest and largest sums for a cage's size can't. These are worth knowing by heart, because each one tells you exactly which digits the cage holds:

Cells	Sum	Digits
2	3	1 2
2	4	1 3
2	16	7 9
2	17	8 9
3	6	1 2 3
3	7	1 2 4
3	23	6 8 9
3	24	7 8 9
4	10	1 2 3 4
4	11	1 2 3 5
4	29	5 7 8 9
4	30	6 7 8 9
5	15	1 2 3 4 5
5	16	1 2 3 4 6
5	34	4 6 7 8 9
5	35	5 6 7 8 9
6	21	1 2 3 4 5 6
6	22	1 2 3 4 5 7
6	38	3 5 6 7 8 9
6	39	4 5 6 7 8 9

Only 8 and 9 can make the 17 cage, so 14 candidates go. From Easy puzzle #1 — play it

Cages that can be made a few ways still narrow things down. A two-cell 10 can't hold a 5, since that would need a second 5. And remember the cage's neighbours: a digit already placed in the same row, column or box rules out every set that needs it there.

Singles, as in Sudoku

Once cages have narrowed things down, the Sudoku rules do the rest. A digit with only one place left in a row, column or box goes there, and a cell with only one digit left takes it.

In box 9, only one cell can still hold a 7. From Easy puzzle #1 — play it

The rule of 45

The digits 1 to 9 add up to 45, so every row, column and box adds up to 45 too. Take a box, and the cages lying wholly inside it. If they cover every cell but one, that cell, an innie, is 45 minus their sums.

The cages wholly inside box 9 add up to 39, so the one cell left over, at row 7, column 7, is 45 − 39 = 6. From Easy puzzle #3 — play it

It works the other way round too. If the cages that cover a row, column or box between them have just one cell outside it, that cell, an outie, is their sums minus 45.

The cages covering box 1 add up to 49, and only one of their cells lies outside it, at row 4, column 1. That cell is 49 − 45 = 4. From Easy puzzle #1 — play it

Make a habit of it: before anything else, run your eye along each row, column and box and ask which cages fit wholly inside.

Medium: two or three cells left over

Often the cages leave two or three cells over instead of one. You don't get a digit straight away, but you get a new sum, and that works just like a small cage — with one difference: two leftover cells that don't share a row, column, box or cage may hold the same digit.

The cages wholly inside row 1 add up to 31, so the three outlined cells add up to 45 − 31 = 14. With what they can still hold, 2 candidates go. From Medium puzzle #1 — play it

Medium: a cage that claims a digit

Sometimes you can't place anything in a cage yet, but every way of making it puts a certain digit in the part of it that lies in one row, column or box. Whichever way the cage is made, that unit gets the digit from the cage, so the rest of the unit can't have it.

Every way to make the 22 cage puts a 9 in row 4. So no other cell there can be a 9, and 2 candidates go. From Medium puzzle #1 — play it

Medium puzzles may also need Sudoku's pointing, claiming and pairs; our Sudoku strategy guide shows each one.

Hard: the rule of 45 across several units

Two rows side by side add up to 90, three to 135; two neighbouring boxes add up to 90 too. The rule of 45 works across them just as well, and because a wider region swallows more whole cages, it often finds a sum that a single row or box can't.

Together, rows 1 and 2 add up to 90. The cages covering them make 100, so the two cells sticking out add up to 100 − 90 = 10. With what they can still hold, 2 candidates go. From Hard puzzle #1 — play it

Hard puzzles may also need Sudoku's triples or an X-Wing, shown in the Sudoku strategy guide.

Good habits

Start with the one-way cages. A 3, a 17 or a three-cell 24 tells you its digits outright.
Look for 45s early. Box by box, row by row: which cages fit wholly inside, and what's left over?
Pencil in pairs. Knowing a two-cell cage is "1 and 2" is often enough to rule both digits out of the rest of its row, column or box.
After each digit, revisit its cage. One placed digit often settles the rest of a small cage at once.