45 Rule — Killer Sudoku Technique

What it is

Because the digits 1–9 sum to 45, every row, column, and 3×3 box in a Killer Sudoku must total 45. When the cages inside a unit are fully contained — none crosses into a neighbouring row, column, or box — their sums together must equal 45. The technique becomes a deduction engine when ONE cell of the unit lies outside the unit's fully-contained cages (i.e. it belongs to a cage that extends into a neighbouring unit). Subtract the sums of the fully-contained cages from 45 and the difference is that cell's value — directly, with no candidate work. The 45 rule is the foundation every other Killer technique builds on.

When to use it

First. Before any other Killer technique. Sweep every row, column, and box for one with a single outside-the-cages cell. That cell's digit is forced by the 45 rule with no further reasoning.

Worked example

Row 1 contains three cages fully inside it summing to 41 (say 15 + 18 + 8) and one extra cell R1C9 that belongs to a 12-cage extending into row 2. The fully-contained cages cover 8 cells totalling 41; the remaining cell R1C9 = 45 − 41 = 4. Place 4 in R1C9 — the cage R1C9 belongs to and its candidates didn't enter the deduction at all.