WHERE 45 COMES FROM
A standard Sudoku row contains 1–9 exactly once. The sum is always 1+2+3+4+5+6+7+8+9 = 45. This is true for every row, every column, and every 3×3 box — 27 regions, all summing to 45.
In regular Sudoku, this fact is a curiosity. In Killer Sudoku, it becomes a weapon.
Killer Sudoku removes all starting digits and replaces them with dashed cages, each labelled with a target sum. The cage sums are your only arithmetic clues. The Rule of 45 lets you derive additional constraints from the structure of the grid itself.
THE BASIC APPLICATION
Suppose a row has four cages entirely within it, summing to 38. The row total must be 45. The fifth cage must therefore contain one cell in this row summing to 45 − 38 = 7.
That single cell is now locked to 7. You didn't scan for naked singles. You didn't need any given digits. You used arithmetic against the known total of the region.
This is the basic form: identify a region (row, column, or box), sum all the cage values that fall entirely within it, and subtract from 45. The result is what the remaining cage cells must contribute to that region.
THE OUTHOUSE TRICK
The real power of the Rule of 45 appears when cages straddle region boundaries. Take a 3×3 box where all cages but one are entirely inside the box. That one cage has cells both inside and outside.
Sum the cages entirely inside the box. The box total is 45. The difference tells you exactly what the stradding cage must contribute from its cells inside the box.
If the straddling cage sums to 14 total and must contribute 5 from its two cells inside the box, then the cage's cells outside the box must sum to 14 − 5 = 9. You've now constrained cells in a different region without scanning them at all.
Experienced Killer solvers often spend the first minute of a puzzle just running 45-rule calculations across every row, column, and box. The constraints this generates make the middle of the puzzle trivial.
MULTI-REGION APPLICATIONS
The Rule of 45 extends naturally to groups of regions. Two adjacent rows contain 90 total (2 × 45). Three rows: 135. This lets you handle puzzles where individual rows don't have neatly contained cages.
A common pattern is the 2-row outhouse: a cage straddles two rows and you can't isolate it in either row alone. But if you sum both rows (90), then sum all cages entirely within those two rows, the remaining cage cells must account for the difference. Often this is enough to pin the straddling cells.
The same logic applies to L-shapes, T-shapes, and arbitrary combinations of boxes and rows. The rule is the same: region total is always a multiple of 45, cage arithmetic fills in the rest.
WHEN THE RULE DOESN'T HELP
The Rule of 45 is most useful when cages align cleanly with region boundaries — when most cages in a row stay in that row, with one or two straddlers. Highly fragmented grids where every cage crosses multiple boundaries give the rule less purchase.
In those cases, unique-sum cages (2-cell cages summing to 3 or 17, 3-cell cages summing to 6 or 24) are your entry point instead. Once you have enough candidates from unique sums, standard Sudoku scanning takes over.
The two approaches are complementary. A well-constructed Killer puzzle typically lets you make progress from both ends — and the 45-rule often breaks open the section that unique sums couldn't reach.