What is the 45 rule in Killer Sudoku?
The 45 rule is the single most powerful technique in Killer Sudoku. Master it and you will unlock free digits in almost every puzzle, regardless of difficulty.
Why Every Unit Sums to 45
Every row, column, and 3×3 box must contain the digits 1 through 9 exactly once. Since 1+2+3+4+5+6+7+8+9 = 45, every complete unit sums to exactly 45. This is a fixed mathematical fact - It cannot be otherwise in any valid Sudoku solution.
How to Apply the Rule
Step 1: Pick a unit (row, column, or box).
Step 2: Add up the sums of all cages lying entirely within that unit.
Step 3: Note any cells whose cages cross the unit boundary.
Step 4: 45 minus the Step 2 total gives the sum of those crossing cells.
If only one cell crosses the boundary, its exact value is 45 minus the cage-sum total. You have a free digit.
Worked Example
Suppose box 1 (top-left 3×3) has cages summing to 38 entirely within it. One additional cell from a cage extends outside box 1. That cell must equal 45 - 38 = 7. Place 7 immediately.
Now suppose the total of entirely-contained cages is 31, and two cells from external cages cross into the box. Those two cells together sum to 45 - 31 = 14. Their candidates are restricted to pairs summing to 14: {5,9}, {6,8}.
Extending Across Multiple Units
The 45 rule can be applied to combinations of units. Two rows together must sum to 90. Three boxes in a band must sum to 135. This extended form can resolve multi-cell innies and outies that span several units, giving you sum constraints even when no single cell is fully determined.