Beginner

The 45 rule

The digits 1 through 9 add up to 45. So every row, every column, and every 3×3 box on a killer sudoku board sums to 45, whether or not you have filled a single cell. Cages are just another way of packaging those same digits. When the packaging does not line up with a house, the difference is a digit.

Take a row that is covered by four cages except for one cell whose cage spills into the row below. Add the cage sums that sit entirely inside the row, and add the parts of the spilling cage that you already know. Whatever is missing from 45 is the value of the leftover cell. You never needed to list combinations. The total did the work.

The same arithmetic runs backwards. Add every cage that touches the row. Their sums count the row plus any cells hanging outside it. Subtract 45 and the overhang is what those outside cells must add up to. One outside cell means you can write the digit immediately. Two outside cells means you have a new little cage whose sum you just discovered, and the combination list takes over from there.

Boxes are usually richer than rows, because cages wander across box borders more often than they wander across a whole row. Look at each box and ask only one question: how many cells of crossing cages sit outside this box? If the answer is one, you have an outie. If the answer is one cell inside the box that belongs to a cage living mostly outside, you have an innie. Both are the 45 rule wearing different clothes.

Two or three houses can be added together. A band of three boxes is 135. Two stacked rows are 90. The bookkeeping is the same: cage sums that cover the region, minus the region total, equals the cells left outside. Larger regions are how hard puzzles hide a 45-rule step inside what looks like a blank middle.

Do the addition twice. A one-digit slip in a cage sum sends you hunting a digit that cannot exist, and the puzzle feels unfair when it is only a arithmetic error. On this site the hint button will name the house and the cell when a 45-rule step is the simplest one available.

Use this technique on a puzzle

Keep the combination table open while you read, and the rules if the cages themselves are still new.