Nurikabe

Nurikabe — the name is a yōkai, a wall that appears in the road at night — asks you to shade a sea around a set of islands. Each number is an island of exactly that many cells, islands never touch side by side, the sea is a single connected region, and the sea never fills a 2×2 square. Those last two rules are what make it hard: they are global constraints, so no amount of purely local reasoning is guaranteed to finish a board.

Boards here are carved by repair. The generator starts from an all-sea grid — which satisfies connectivity for free and breaks the 2×2 rule everywhere — then converts cells to island one at a time, never merging two islands and never cutting the sea in half. Every candidate is then checked to have exactly one solution and stamped with the deepest rule that solution actually needs: openers, reachability, connectivity, or search.

The maths behind it, at length: Nurikabe, and what replaces candidate enumeration.