Why does every Sudoku row total 45?
Each Sudoku row, column, and 3×3 block contains the digits 1 through 9 exactly once. The sum 1+2+3+4+5+6+7+8+9 is 45. Killer Sudoku turns this fact into a sum-solving tool when cages cover most of a region.
The Rule of 45 does not replace Sudoku rules; it only adds another constraint. After finding the total of an unknown cell or group, you still need row, column, and block candidates to determine the exact digits.
Cages fully contained within the region
If cages entirely inside a block total 38 and only one cell remains outside those cages, that cell must be 45−38=7. If two cells remain, you know their combined total but still need candidates to select a valid pair.
Check carefully that the cage is fully contained in the region. If a cage crosses a block boundary, adding the full cage total also counts outside cells and produces the wrong result.
- Identify the exact row, column, or block boundary being calculated.
- Only add a whole cage if every cell in that cage lies inside the region.
- Subtract the known total from 45 to find the remainder.
Handle cages crossing a boundary
When a cage crosses a region boundary, compare the cells inside the region (innies) with those extending outside (outies). The region itself must still total 45, so the difference between cage totals and 45 reveals the sum of the excess or missing cells.
Do not try to calculate a complex shape in your head immediately. Mark the group of cells being summed, write the sum expression, then simplify it. A clear calculation is more reliable than remembering an “outie formula” without checking the boundary.
A correct total does not mean the digits are correct
If two cells total 10, several non-repeating pairs are possible, such as {1,9}, {2,8}, {3,7}, and {4,6}. You cannot choose a pair from the sum alone. Use the related rows, columns, blocks, and cages to eliminate invalid pairs.
The Rule of 45 is strongest when combined with candidates. After each sum calculation, turn that information into a valid candidate set, then return to Classic Sudoku logic.
Apply it immediately on a real puzzle
A technique becomes automatic only after you use it across many different boards. Play at a manageable level, prioritize certain deductions, and review the step that caused you to get stuck.