Cross-unit 45 rule
The 45 rule applied across two or more units that a cage spans. Sums and the cells that cross unit boundaries balance via the same 45-each-unit constraint.
The cross-unit 45 rule is the broader form of the 45 rule and its innies-and-outies cousin. Each row, column, and box on a Sudoku grid sums to 45. When a cage spans two or more units, the cage's cells contribute to the sum of each unit it touches; the 45-rule arithmetic can be applied to each unit independently, and the constraints between them often pin cage cells into tighter combinations than any single-unit application would.
The simple two-unit case
A cage with 3 cells in row 4 and 2 cells in row 5, with sum 28. Row 4's nine cells sum to 45; row 5's nine cells sum to 45. The cage's row 4 cells sum to 45 minus the rest of row 4; the cage's row 5 cells sum to 45 minus the rest of row 5. The two derived sums must add to the cage's total (28), which constrains them simultaneously.
If row 4 outside the cage contains 1, 3, 5, 7, 8, 9 (already known and totalling 33), the cage's row 4 cells sum to 45 - 33 = 12. The cage's row 5 cells then sum to 28 - 12 = 16, with whatever digits remain available in row 5. Now the cage's two halves are pinned to specific sums, and the unique-combinations reasoning fires on each half independently.
The same reasoning generalises across any number of units a cage crosses. A 6-cell cage crossing two rows and two boxes admits four independent 45-rule applications, each producing a derived sum that the others must reconcile with.
Where it fits between the named killer techniques
The cross-unit 45 rule is the bridge between innies-and-outies and the more advanced cage-overlap reasoning. Innies-and-outies treats the cells of partial cages as innies or outies of a single unit; the cross-unit 45 rule treats the same cage as part of two units simultaneously, balancing the sums across both. Cage overlap pushes the same logic further by considering multiple cages in the same intersection.
In practice the three techniques blur together. Many expert killer solvers don't distinguish them by name; they apply 45-rule arithmetic at whatever level the puzzle's geometry demands, naming only the move that breaks the puzzle.
When you'll see it
Cross-unit 45-rule moves appear constantly on hard and expert killer puzzles where cages span multiple units. The signature is a stuck cage that has too many possible combinations to enumerate but whose cells touch a row, column, or box where the rest of the digits are mostly known. Working out the derived sum on the partly-known unit usually pins the cage's contribution into a narrow band, and a killer pair or unique combinations move follows.
See also
- 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.
- 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.
- 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 overlap— Killer-Sudoku reasoning across cages that share cells with units. The shared cells must satisfy both the cage's sum and the unit's 45 rule, surfacing eliminations.
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.