Reach and room

Big regions are the hardest part of Fillomino. Two simple measurements, how far a region can reach and how much room it has, place most of them for you.

Reach

A region of size N can extend at most N − 1 steps from any of its cells, counting steps through shared sides. So a 3-clue can only claim cells within two steps of it; a 2-clue only its direct neighbours.

A 3 can reach only the shaded cells. Anything farther away belongs to some other region.

Reach is most useful for ruling things out. If an empty cell is out of reach of every clue with a given number, that cell can't be part of any of those clued regions. If it's out of reach of every clue, it must belong to a region with no clue at all, which makes it a candidate for the reasoning in Enclosed areas.

When two clues of the same number are far apart, reach also tells you whether they could be in the same region. Two 4-clues that are five steps apart can't share a region, because a region of four cells spans at most three steps.

Room

The flip side of reach is room. A region needs space to reach its size, and walls, edges and finished regions all take space away. Count the empty cells a clue could possibly grow into. If the count is exactly the number it needs, every one of those cells is part of the region.

The 5 is boxed in by the board's top edge, a 1 and a finished row of 4s.

Exactly five cells are available, so all of them are 5s.

This is the same idea as the only way out in First moves, scaled up. A 5 with exactly five cells of room has no choices left. With six cells of room, one cell is left over; often you can't tell which yet, but cells that every possible shape must include are already certain. Those are usually the cells nearest the clue, and the narrow necks the region has to pass through.

Squeezing between two clues

Room interacts with rule 2. Say two 4-clues sit in the same row with three empty cells between them. Joining them would take at least five cells, so they must be separate regions, and separate regions of 4s can never share a side. That means the gap can't be filled with 4s all the way across: at least one of those three cells holds a different number and acts as a wall between them. Often the rest of the board decides which cell that is. On medium and hard boards, many of the decisive steps are exactly this: two big regions of the same size competing for space, where only one arrangement keeps them apart.