Killer Sudoku Cage Combinations & the 45 Rule
Killer Sudoku swaps most given digits for cages — dotted regions that must add up to a printed sum, with no repeated digit inside. Fast solvers rarely add cell by cell. Instead they memorize a short list of cage combinations and let arithmetic do the eliminating. Here is that list, plus the 45 rule that ties it all together.
Every row, column, and box sums to 45. Killer Sudoku is arithmetic wearing a logic mask.
What are cage combinations?
A cage combination is the set of digits that can legally fill a cage. A two-cell cage summing to 3 can only be 1+2; a sum of 17 can only be 8+9. A 16 cage in two cells must be 7+9, because 8+8 would repeat a digit. Extreme sums — the smallest and largest possible for a cage size — allow the fewest combinations, so they are always your best entry points: two cells with 3, 4, 16, or 17; three cells with 6, 7, 23, or 24.
How to crack cages with the 45 rule
- Pencil in the unique cages first. Cages with only one possible digit set (like a two-cell 17 or a three-cell 24) hand you candidates for free. Mark them before anything else.
- Apply the 45 rule. Every row, column, and 3×3 box sums to 45. If cages exactly cover a box except for one cell, subtract their sums from 45 — the remainder is that cell’s value.
- Cross-check with classic rules. Cage candidates still obey row, column, and box constraints. A 7+9 cage crossing a column that already contains a 9 collapses to a single arrangement.
- Recalculate big cages as digits land. Once one cell of a large cage is fixed, subtract it from the target and treat the rest as a smaller cage — its combination list shrinks every time.
Diagram: a two-cell 16 cage can only hold 7 and 9.
Drill these combinations until they feel like reflexes, then put them under pressure: Hell Mode on Sudoku Hot is full Killer Sudoku across five tiers, and the daily challenge rotates cage layouts so the arithmetic never goes stale.
