Partial cage combinations
Killer-Sudoku reasoning where partial information about a cage's digits — eliminations, placements, or external constraints — narrows the cage's combinations to a smaller set.
Partial cage combinations is the killer Sudoku move that runs when a cage's possible digit combinations have been narrowed by external information — a placement elsewhere, an elimination forced by a unit constraint, a digit ruled out by another cage. The narrowed combination set is smaller than the cage's full set, and the smaller set often forces eliminations across the cage's cells that the full enumeration didn't.
The basic shape
A 3-cell cage with sum 18 has possible combinations {1, 8, 9}, {2, 7, 9}, {3, 6, 9}, {3, 7, 8}, {4, 5, 9}, {4, 6, 8}, {5, 6, 7}. Suppose 9 has been placed somewhere in a unit shared by two of the cage's cells, eliminating 9 from those cells. The combinations that include 9 in only the third cell remain valid; the combinations that need 9 in either of the first two cells are eliminated. The remaining combinations are narrower, and the digits available to each cage cell shrink accordingly.
Working through the same example: {3, 7, 8} and {5, 6, 7} survive (neither needs 9 in cells 1 or 2); the others go. The cage's digit set is now {3, 5, 6, 7, 8}, down from the original {1, 3, 4, 5, 6, 7, 8, 9}. Many cells in the cage and its unit constraints can now be sharpened.
The move is mechanical once the partial information is in hand. The discipline is in maintaining the cage's combination set as candidates change elsewhere — most expert killer solvers track active cage combinations as a side calculation and update them whenever a placement or elimination touches a cage cell.
Why it's classified intermediate
Partial cage combinations is a more accessible move than the rest of the 4.4 killer cluster. The reasoning is direct: known information narrows known possibilities. There's no chain to construct, no fish to spot, no uniqueness argument to make. The technique earns its place in the glossary because the practice of maintaining the combination set is non-trivial, and because the same partial-information principle underlies more advanced killer techniques like cage overlap and sum chain.
When you'll see it
Constantly, on any killer puzzle past easy. Partial cage combinations is less a discrete move than a continuous bookkeeping exercise: every placement, every elimination, every external constraint feeds back into cage combination sets. Solvers who maintain the bookkeeping fluently solve killer puzzles substantially faster than those who recompute combinations from scratch each time.
The full reference for cage combinations — the canonical sum × cell-count → digit-sets table — lives at unique combinations for fully forced cages, but the partial-information variant covered here is the more common practical move.
See also
- Unique combinations— In Killer Sudoku, cage sums whose cell count and total leave only one possible digit set. The arithmetic shortcut behind most killer pair and triple deductions.
- 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.
- 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.
- Killer triple— In Killer Sudoku, when three cells in the same unit are confined to the same three-digit set 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.