The 45 rule
In Killer Sudoku, the fact that every row, column, and 3×3 box must sum to 45 — because 1+2+…+9 = 45. The foundational arithmetic identity behind most killer techniques.
The 45 rule is the foundational identity behind almost every Killer Sudoku technique. It's a one-line consequence of the standard Sudoku no-repeats rule: every unit — every row, column, and 3×3 box — contains each of the digits 1 through 9 exactly once. The sum of those nine digits is 1 + 2 + ... + 9 = 45. So every row, every column, and every box sums to 45, no matter what cage configuration the puzzle uses.
Why it matters
In Classic Sudoku, the no-repeats rule is the constraint and the 45 sum is a curiosity. In Killer Sudoku, the cage sums are the constraint, and the 45 sum becomes a tool. If you know what the cages inside a row sum to, you know what the cells on that row's boundary must sum to, because the row total is locked at 45. Most killer-specific deductions are some version of this arithmetic.
How it gets used
The classic application is innies and outies. Take a row whose cages all fit cleanly inside the row except for one cell that "spills over" into a different row. Sum the cage totals — the spillover cell's value is whatever makes the row sum to 45. The rule turns into a placement.
The same trick works for columns and boxes. A box whose cages all fit inside it sums to 45 internally; a box with a spillover cell needs the spillover to fill the gap. A column with two cages partially inside it can be reasoned about by subtraction.
Why directional variants exist
Sudoku Mountain's solver internally tracks three flavours of the rule, one for each unit shape (row, column, box), because the same identity applies along each. The mathematical claim is identical; only the unit type changes. The split lets the solver replay name which direction it used.
For an introduction to Killer Sudoku as a whole and where the 45 rule fits in the broader technique set, see Meet Killer Sudoku. The closely-related deduction shape that sits one step deeper is innies and outies.
See also
- Cage— In Killer Sudoku, a contiguous group of cells outlined by a dotted line, with a printed sum the digits inside must add up to. Replaces the classic Sudoku given.
- Cage completion— In Killer Sudoku, placing the last digit of a cage by subtracting the digits already in it from the cage's sum. The cage's leftover arithmetic does the work.
- Innies and outies— In Killer Sudoku, deducing a cell's digit by applying the 45 rule to a unit whose cages partly overlap with — or partly spill out of — that unit.
- Killer pair— In Killer Sudoku, when two cells in the same unit are confined to the same two-digit pair by their cage's arithmetic — eliminating those digits from elsewhere in the unit.
Read more
- Meet Killer Sudoku
An introduction to Killer Sudoku for someone who knows the classic version — what changes, how the experience differs, and where to start.