Unique combinations

A unique combination is a cage whose cell count and sum leave only one possible digit set. The cage's arithmetic does most of the work — once you've identified which digits the cage must contain, the remaining puzzle is "in what order do those digits sit inside the cage" rather than "what are the digits at all."

The standard list

A handful of cage shapes have unique combinations because there's only one way to write the sum as a set of distinct digits 1-9. The most useful ones:

• 2-cell cages: sum 3 → {1, 2}; sum 4 → {1, 3}; sum 16 → {7, 9}; sum 17 → {8, 9}
• 3-cell cages: sum 6 → {1, 2, 3}; sum 7 → {1, 2, 4}; sum 23 → {6, 8, 9}; sum 24 → {7, 8, 9}
• 4-cell cages: sum 10 → {1, 2, 3, 4}; sum 11 → {1, 2, 3, 5}; sum 29 → {5, 7, 8, 9}; sum 30 → {6, 7, 8, 9}
• 5-cell cages: sum 15 → {1, 2, 3, 4, 5}; sum 35 → {5, 6, 7, 8, 9}

The pattern: extreme sums (close to the minimum or maximum possible for a cage of that size) have fewer arrangements, often just one.

How they're used

Unique-combination cages are entry points. Spotting one tells you exactly which digits are in the cage; combining that with the cells' row, column, and box constraints often forces a placement immediately. They're the killer-Sudoku equivalent of "look for the easiest naked single first" — find the cage with the most-constrained sum, work outward.

The big strategic move that flows from unique combinations is the killer pair — when a unique-combination cage sits inside a single unit, its digits are locked to that unit's cells, eliminating them from the unit's other cells.

For an introduction to Killer Sudoku and where unique combinations fit in the broader technique set, see Meet Killer Sudoku.