Cage uniqueness

Cage uniqueness is killer Sudoku's analogue to unique rectangle reasoning. Some cage configurations, if they were allowed to complete in multiple ways, would produce two valid puzzle solutions. The puzzle's uniqueness guarantee rules out those configurations, so any move that would lead to a multi-solution cage state is forbidden — and the move that prevents it is forced.

The basic shape

Consider a 2-cell cage with sum 10. The valid digit combinations are {1, 9}, {2, 8}, {3, 7}, and {4, 6}. Each combination is a pair of digits that can occupy the cage in two orders. If the cells of the cage share a unit (which they often do — most 2-cell cages live within a row, column, or box), only one of those orderings will be valid given the rest of the unit's constraints.

The cage-uniqueness move fires when a cage's pair of cells is currently constrained to exactly one digit pair, and both orderings of that pair appear consistent with the rest of the puzzle. If both orderings would close the puzzle to a valid solution, the puzzle would have two solutions — which the uniqueness guarantee forbids. So one of the orderings must be inconsistent with some constraint we haven't yet applied, and the technique points us toward where to look for that constraint.

In practice, cage uniqueness most commonly fires when a cage's combination overlaps with a unique rectangle-like structure across the rest of the grid: the cage's two cells share a "deadly pair" relationship with two cells outside the cage. The cage-uniqueness move resolves which orientation the cage takes, breaking the deadly pattern.

Larger cages

Larger cages admit cage-uniqueness reasoning when their possible digit combinations include a pair whose two orderings both leave the rest of the puzzle resolvable. The argument is structurally the same: the puzzle's uniqueness rules out the two-solution case, forcing a particular ordering or eliminating a particular combination.

These configurations are rarer on larger cages because the combinatorial space of possible digit assignments is larger and the conditions for "exactly two valid solutions" are restrictive. When they appear, the move is decisive — usually placing one of the cage's digits and unblocking the surrounding unit immediately.

When you'll see it

Cage uniqueness shares the philosophical reception of unique rectangle: some solvers refuse to use it on the grounds that "the puzzle is unique" is a meta-property rather than a no-repeats consequence; others accept it because every published killer Sudoku has been checked for uniqueness. In Sudoku Mountain's killer puzzles, the move is safe to use.

The signature is a stuck state late in the puzzle where most cages are mostly resolved, but two cells in a small cage still have two candidates each that could swap. Check whether the swap would produce a valid alternate solution. If yes, the move is alive.