Killer Sudoku cage combinations

A cage is a group of adjacent cells outlined by a dashed border. The small number in the corner is the target sum. The cells inside the cage must add up to that sum, with no digit repeated inside the cage.

Cages reduce a Sudoku problem to two simultaneous questions: which digits can fill these cells, and how can they sum to the target. The digit-set question is what these tables answer. The placement question is what classical Sudoku techniques solve.

A cage of six or more cells has too many combinations to list usefully. Don't enumerate the cage; enumerate the cells around it.

If a six-cell cage sits inside a single 3x3 box, the three cells of the box outside the cage form an implicit three-cell cage with a known sum: 45 minus the original cage's sum. Read the small cage's combinations from the three-cell table and the original cage holds the complementary digits.

The same trick works for seven-cell cages (the complement is two cells, read from the two-cell table) and eight-cell cages (the complement is a single cell).

Every row is a target sum. Every entry is a possible digit set, written without separators (12 means {1, 2}; 1234 means {1, 2, 3, 4}).

The smallest possible sum for an N-cell cage is 1+2+...+N. The largest is (10-N)+(11-N)+...+9. Sums outside that range cannot exist.

If the cage's house already contains certain digits, you can cross out any combination that includes those digits. What survives is the actual set of possibilities, and often only one combination remains.