The Rule of 45
The Rule of 45 is the single most important idea in Killer Sudoku. It comes from one plain fact: the digits 1 to 9 add up to 45. Since every row, column and 3×3 box contains each digit exactly once, every one of them sums to 45. In Killer, where cages give you partial sums, that lets you read a hidden cell straight off the arithmetic.
How the rule works
Take any single unit — say a column. Suppose the cages that lie entirely inside it add up to 38, and one more cell of the column belongs to a cage that pokes outside. Because the whole column must total 45, that one remaining cell has to be 45 − 38 = 7. You've placed a digit with pure addition, no candidates required.
The rule scales up. If the inside cages total 40 and two cells stick out, those two cells sum to 5 — which sharply limits them to {1,4} or {2,3}.
A worked example
Look at the bottom row of a Killer grid. Three cages sit fully inside it with sums 12, 9 and 17 — that's 38. A fourth cage overlaps the row with just one of its cells inside. The row must total 45, so:
45 − (12 + 9 + 17) = 45 − 38 = 7
That overlapping cell is a 7. Everything else in its cage now has to work around that.
The bottom row: three cages lie fully inside it — blue 12, pink 9, green 17, totalling 38. The row must sum to 45, so the one cell poking out of a fourth cage (amber) is 45 − 38 = 7.
From one unit to many
The same trick works across blocks of units. Two stacked boxes total 90; three total 135. Add up the cages fully inside a block, compare with that total, and the cells poking in or out are pinned down. That extension has its own name — innies and outies — and it's where the rule of 45 really earns its keep.
Common mistakes
- Miscounting the boundary. Only cages fully inside the unit count toward the inside total; the poking cage is what you're solving for.
- Forgetting it's per unit. Each row, each column and each box totals 45 independently — pick the one with the cleanest cage boundaries.
- Stopping at single cells. When two or three cells stick out, you still learn their sum, which is often enough to crack the cage.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.