Innies and outies

Innies and outies is the killer-Sudoku move that puts the 45 rule to work on units whose cages don't sit cleanly inside the unit's boundary. The rule says every row, column, and 3×3 box sums to 45; when you can compute what the cages contribute to that 45, the leftover cell — the "innie" or the "outie" — is fixed by subtraction.

The innie

An innie is a single cell of a unit that isn't covered by any cage that sits entirely within the unit. The cages that do sit entirely within the unit cover 8 cells; the innie is the 9th. Sum the cage totals — the innie's value is 45 minus that sum.

Worked example: row 5 contains three cages, all entirely inside row 5, summing to 7 + 14 + 21 = 42. There's one cell left in the row (the innie), and its value must be 45 − 42 = 3. Place a 3.

The outie

An outie is the mirror move. A cage sits mostly inside a unit but has one cell that spills out into a neighbouring row, column, or box. The cages-that-fit-cleanly contribute their sums to the unit; the outie's cage contributes its sum minus the spillover cell. Setting the unit total to 45 and solving for the spillover cell gives a placement.

The arithmetic for outies is slightly more involved than for innies because you're working with a partial cage sum, but the pattern is the same: the 45 rule turns a unit's cage configuration into a placement.

When you'll see it

Innies and outies are the workhorse killer-Sudoku technique past the cage single / cage completion layer. Hard and expert killer puzzles often require multiple consecutive innies-and-outies arguments before any cell becomes a naked single. Once you're comfortable with the 45 sum, scanning for them becomes second nature — every row, column, and box is a candidate for the move.

For an introduction to Killer Sudoku and where innies and outies fit in the broader technique set, see Meet Killer Sudoku.