How to solve Killer Sudoku: cage sums and the rule of 45

To solve a Killer Sudoku, start with the cages whose total can only be made one way, then use the fact that every row, column and box adds up to 45 to find the squares the cages leave out. The usual sudoku rules do the rest. Every Gridarium killer sudoku can be solved this way, without guessing.

How to read the cages

The grid follows the sudoku rules: the digits 1 to 9 once in every row, column and 3 by 3 box. On top of that, dotted lines group the squares into cages. The small number in a cage's corner is the total of its digits, and a digit never repeats inside a cage. Few or no digits are given: the totals are the clues.

Step 1: the cages with only one answer

Some totals can only be made one way with different digits. Learn these and you will spot a starting point in almost every puzzle: the digits are known, only their order is left to find.

Step 2: count the ways to make a total

Most totals can be made in several ways, but fewer than you might think, because the digits in a cage must all differ. Two squares adding up to 10 can only be 1 + 9, 2 + 8, 3 + 7, 4 + 6: 4 ways, and never 5 + 5. Write the possible digits as notes, then cross them out as the rows, columns and boxes around the cage fill in.

Step 3: the rule of 45

The digits 1 to 9 add up to 45, so every row, column and box adds up to 45. When the cages that sit wholly inside a box cover all its squares but one, that last square is 45 minus their totals. The same works for a row or a column, and for two rows side by side (90).

An example from the daily puzzle of Tuesday, October 20: in box 2, 3 cages sit wholly inside the box, with totals 19 + 11 + 13, which makes 43. The only square of the box they do not cover, row 1, column 6, is therefore 45 − 43 = 2.

Use the Cage sums button

In Gridarium, select a square and press Cage sums: you see every set of digits its cage can still hold, given what is already written in it. It explains instead of answering, so it never counts as a hint. If no set fits, one of the cage's digits is wrong.

The three levels

• Easy: small cages, at most 4 squares, and up to 10 digits given. Cage arithmetic is all you need.
• Medium: cages of up to 4 squares and at most 4 digits given. You will need the rule of 45 at least once.
• Hard: cages of up to 5 squares and no digit given. The rule of 45 is needed, and sometimes the classic sudoku eliminations too (pairs, locked candidates).