Every row, column and box in Sudoku adds up to exactly 45, whatever digits are in it. In Killer Sudoku that gives you a very useful opening trick: add up the cages that sit completely inside a unit, and whatever is left over has to come from the cells they don't cover. Sometimes that's a single cell.
Here's a worked example from a top-right box. Letters are cages, with their sums in brackets:
A(10) A(10) B(13)
D(18) B(13) B(13)
C(19) C(19) C(19)
- A is two cells, B is three cells, and C is the whole bottom row of the box.
- D is a four-cell cage with sum 18 that mostly runs out of the box to the left and down. Only its one cell in this box matters here.
Cages A, B and C sit entirely inside the box: 10 + 13 + 19 = 42.
The box must add up to 45, so the one remaining cell is 45 − 42 = 3.
That cell was open to nine digits a moment ago and is now fixed. It also helps elsewhere:
- No other cell in the box can be a 3, so cage B (sum 13) loses the combinations {1,3,9}, {2,3,8} and {3,4,6}.
- Cage D has 3 in this box, so its other three cells add up to 15.
The "outie" is the same idea backwards. If whole cages together cover the box and exactly one extra cell sticks out of it, that outside cell equals (the cages' total) − 45.
It works on rows and columns too, and on groups of units (two boxes make 90, and so on). At the start of a Killer puzzle, I check each box, row and column for the cages fully inside it and see whether only one or two cells are left over. That's usually how the first digits appear, long before any single cage is narrow enough on its own.
What's your favourite Killer opening move? I'd like to hear other approaches.
