What is the 45 Rule in Sudoku?

The 45 rule in Sudoku is a basic solving strategy that comes from the fact that the digits in each row, column and 3×3 box will always sum to 45. 

This is because each of these regions must contain the digits 1 to 9 exactly once, and when you add all these digits up, you get 45. 

Knowing about the rule of 45 can be useful when playing Sudoku variants where different rule sets can mean knowing the value of a set of digits sum to can help eliminate candidates. 

This is particularly true when solving Killer Sudoku puzzles. 

In Killer Sudoku, cages are overlaid over the regular Sudoku grid and the sum of the cages is given in the top left corner. Such as in the example below. 

The bottom right-hand cage is labeled 19, so the four cells that comprise this cage must sum to 19. 

What you’ll also notice is that there are two other cages contained entirely within the bottom right 3×3 box. Together with the 19 cage, these three cages sum to 39 (19+12+8=39). 

Because there is only one additional cell in this 3×3 box that isn’t covered by these three cages, and we know about the 45 rule. The only way to make the 3×3 box sum to 45 is to place a 6 in the remaining cell. 

Remember, every row, column, and 3×3 box in a completed Sudoku puzzle must always sum to 45. Regardless of which combinations of numbers are used.  

In addition to Killer Sudoku, knowing about Sudoku’s 45 rule can potentially be helpful as a solving technique in other variants where the digit sums are important. Such as Sandwich Sudoku.