Fillomino

Other published titles for the puzzle include Allied Occupation[citation needed].

(When published as Allied Occupation in the World Puzzle Championship, the cells of the grid are circular, but this is purely an aesthetic concern.)

It is common practice in solving a Fillomino puzzle to add numbers to the empty cells when it is determined what size polyomino each must belong to; these numbers are effectively treated identically to the givens.

(Nikoli always publishes solutions to their Fillomino puzzles with both the polyomino borders drawn in and numbers given in every cell.)

From there, the solver searches for three things, possibly in combination: Fillomino adapts to different geometries; hexagonal grids can be used, with the only change in the rules being replacing all instances of polyomino with polyhex.