Cage overlap

Cage overlap is the killer-Sudoku reasoning that fires when a cage's cells partially fall inside a single unit (a row, column, or box) and partially outside it. The cage's sum constraint, the unit's 45 rule, and the cage's unique-combinations constraints all apply to the overlapping cells simultaneously, and the constraint stack often forces eliminations that none of the three alone would reach.

The basic shape

Suppose a cage of 5 cells with sum 27 has 3 cells inside row 4 and 2 cells outside it. Row 4's digits must sum to 45 (the 45 rule); the cage's three in-row cells must sum to some value S that fits both the cage total (27 - sum-of-2-outside-cells) and the row's available digits (the digits not yet placed elsewhere in row 4). Solving the simultaneous constraints often pins down the in-row cells' contribution within a tighter range than either constraint alone permits.

In practice, cage overlap is a fingertip move once you internalise the principle. Most expert solvers run it implicitly — "this cage's cells in box 7 must total at least X and at most Y, given the rest of box 7" — without naming the technique. The named version helps when the constraint chain runs three or four cages deep and the calculation needs to be traced explicitly.

Why it's distinct from innies-and-outies

Innies and outies is the 45-rule applied across an entire unit's boundary: the cells of cages that partially enter the unit are treated as innies (in-unit cells of partly-out cages) or outies (out-of-unit cells of partly-in cages), and the totals balance via the 45 rule. Cage overlap is a more general framing: any cage's overlap with any unit, and the resulting interaction between the cage's sum and the unit's constraints.

Innies-and-outies is the special case where the cages are categorised cleanly into in-cells and out-cells; cage overlap covers the broader space where multiple cages overlap a unit and their sums must be reconciled.

When you'll see it

Cage overlap appears most often on hard and expert killer puzzles where simpler 45-rule moves have already fired. The signature is a cage with 4-6 cells spanning two boxes or a row-and-box intersection, where the sum is far from a tightly forced split. Working out the in-region versus out-of-region totals usually pins one or two cells into a narrower combination, which then surfaces a killer pair or unique combination to finish the move.