Enclosed areas

Some empty cells can't be reached by any clue. They still have to belong to a region, and working out which one is among the most satisfying moves in Fillomino.

Cells no clue can claim

When regions around an area are complete, they form walls. An empty cell, or a group of empty cells, that is sealed in by walls and the edge of the board is an enclosed area. None of the surrounding regions can grow into it, because they're already full. So the area has to be divided into brand-new regions that contain no clues at all.

That sounds like it gives you freedom, but it usually doesn't. A new region can't have the same size as any region it touches along a side, and it has to fit exactly inside the space available.

A single enclosed cell

The highlighted cell touches a region of 3s above, a 2-region to its left, a 2×2 square of 4s to its right and another region of 3s below. Every one of them is complete. If the cell took any of those numbers it would join that region and overfill it. If it took some other number, say 5, it would need four more 5s next to it, and there's no room. The only number that fits a region of exactly one cell is 1, and none of its neighbours is a 1, so the answer is 1.

The reverse matters too. If a single enclosed cell has a 1 as one of its neighbours, it can't be a 1 either, and the position is impossible. That tells you an earlier step was wrong, which is useful when you are checking your work.

Two enclosed cells

A pocket of two cells has only two ways to be filled: a domino of 2s, or two separate 1s. Two 1s side by side would form a region of two 1s, which is illegal. So a two-cell pocket is always a domino of 2s, unless one of its neighbours is a 2-region, in which case something earlier is wrong.

Larger enclosed areas

With three or more cells, more splits become possible: a three-cell pocket could be a single region of 3s, or a 2 and a 1. To choose, look at the neighbours. If a region of 3s borders the pocket, the pocket can't be one 3-region, so it must split. Then the 1 has to go somewhere it doesn't touch another 1, and the 2 somewhere it doesn't touch another 2. Usually only one arrangement survives.

On large boards, enclosed areas often appear late, after you've completed most of the clued regions. That's why it pays to keep the walls in mind: whenever you finish a region, look at the empty cells it borders and ask whether they're now sealed in.