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The 45 Rule in Killer Sudoku — Innies, Outies, and a Worked Example

Every row, column and 3×3 box in a killer sudoku adds up to 45. This one fact turns cage sums into placed digits before any candidate work begins — here is why it holds, how innies and outies fall out of it, and a worked example you can check line by line.
Published 2026.07.31 · 4 min 한국어판 ↗

Killer sudoku gives you few digits to start from — grids often open with no givens at all, nothing but dotted cages and their little sums. The 45 rule is what makes that opening solvable: every row, every column and every 3×3 box must add up to exactly 45, because each contains the digits 1 to 9 once each, and 1+2+3+4+5+6+7+8+9 = 45. Compare cage totals against that fixed 45 and the puzzle starts handing you digits before you have written a single pencil mark.

Why every unit adds up to 45

Classic sudoku’s rule — each row, column and box holds 1–9 exactly once — is a rule about which digits appear, but it quietly fixes their sum too. A unit that contains each digit once cannot sum to anything other than 45, and this holds in every valid puzzle, whatever the cage layout.

Killer sudoku adds cages on top: each dotted region shows the total of the digits inside it, and a cage may not repeat a digit. Cage totals are the puzzle’s given information; the 45s come free with the classic rule. All the techniques in this article come from setting one against the other.

Innies and outies — the single-cell shortcut

Pick a unit — a box is usually easiest — and start adding up cages: the ones that lie wholly inside it, plus any cage that leaves exactly one cell outside. Two configurations come up constantly:

  • Outie. The cages cover the whole unit plus one cell that pokes outside it. The unit itself accounts for exactly 45 of the total, so whatever is left over is the value of that outside cell: outie = cage total − 45.
  • Innie. The cages sit entirely inside the unit but leave one cell uncovered. The covered cells and the missing cell must together make 45, so: innie = 45 − cage total.

Both deductions place a definite digit — no candidates, no case analysis. That is rare currency this early in a puzzle, which is why it is worth scanning every box for innies and outies before anything else.

A worked example, start to finish

Here is the top-left corner of a puzzle built for this article. Three cages touch box 1: cage A runs along the top row and continues one cell past the box border; cages B and C each fill a row of the box. (Letters mark which cage each cell belongs to — the gap in the drawing is the box border, not a break in cage A, which is contiguous.)

        box 1        outside
    +---+---+---+   +---+
    | A | A | A |   | A |    cage A = 16
    +---+---+---+   +---+
    | B | B | B |            cage B = 19
    +---+---+---+
    | C | C | C |            cage C = 15
    +---+---+---+

Add the three cages: 16 + 19 + 15 = 50. Between them they cover box 1 — which must account for exactly 45 — plus the one cell of cage A that sticks out. So that cell is 50 − 45 = 5, written down before any other reasoning has happened.

The solved corner confirms it:

    +---+---+---+   +---+
    | 2 | 1 | 8 |   | 5 |
    +---+---+---+   +---+
    | 9 | 7 | 3 |
    +---+---+---+
    | 4 | 5 | 6 |
    +---+---+---+

Box 1’s nine digits (2+1+8+9+7+3+4+5+6) come to 45, as they must, and the overshoot of 5 is exactly the outie.

The opposite corner of the same puzzle shows the innie version. In the bottom-right box, a 4-cell cage totalling 19 (D) and another totalling 25 (E) sit entirely inside the box and cover eight of its nine cells:

    +---+---+---+
    | D | D | D |    cage D = 19
    +---+---+---+
    | E | E | D |    cage E = 25
    +---+---+---+
    | E | E | ? |
    +---+---+---+

45 − 19 − 25 = 1, so the remaining corner cell is a 1 — an innie, found with the same arithmetic pointed the other way. (Solved, the box reads 3 4 7 / 6 2 5 / 8 9 1: cage D is 3+4+7+5 = 19, cage E is 6+2+8+9 = 25, and the whole box makes 45.)

Stretching the rule: 90, 135, and beyond

Nothing limits the rule to a single unit. Two complete rows must sum to 2 × 45 = 90, three complete boxes to 135, and in general any region assembled from N full units sums to 45 × N, provided the units do not overlap. This matters because cages are awkward shapes: a cage that straddles the boundary of one box often sits entirely inside a stack of two or three boxes. When no single unit gives you a clean innie or outie, widen the region until the cages fit, and the same one-cell arithmetic frequently reappears at the larger scale.

A useful habit: after every few placements, re-scan the boxes nearest your new digits. Cage totals never change, but as cells resolve, the remaining sums shrink, and innies and outies keep being created all the way through the endgame.

Where the rule fits in a real solve

Two cautions keep the arithmetic honest. First, only whole cages count: if a cage is partly inside your region and partly outside, you cannot use its printed total for that region — either widen the region until the cage fits, or leave it out of the sum. Second, the deduction gives you one cell’s value, not a licence to guess neighbouring cells; write the digit and return to normal reasoning.

From there, the 45 rule hands over to cage-combination thinking — a 2-cell cage totalling 3 can only be 1+2, a 3-cell cage totalling 24 can only be 7+8+9 — and the two techniques feed each other: each placed innie or outie shrinks a cage’s remaining total, which narrows its combinations, which resolves more cells, which exposes the next innie. That loop, more than any single trick, is what carries a killer sudoku from an empty grid to a finished one.

The fastest way to make the rule stick is to use it once. Open a killer sudoku and, before touching anything else, add up the cages around one box and see what falls out.

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