Fill every empty cell with a positive number.
Cells with the same number that touch horizontally or vertically form one region.
A region containing number N must contain exactly N cells.
Separate regions of the same size may not touch horizontally or vertically; diagonal contact is allowed.
The puzzle is solved only when every cell is filled and every region has its exact size. Every published Fillomino puzzle has exactly one valid solution.
A cell holding 1 is a whole region on its own, so it can never grow. None of the four cells beside it may be a 1 either — a second 1 touching it would make a two-cell region of 1s, which is impossible.
Example: A clue reads 1. Rule out 1 in the cells above, below, left and right of it before you go any further.
When a block of equal numbers already has exactly as many cells as its number, it is done. Every cell touching it must hold something different, and the boundary the board draws around it is permanent.
Example: Three 3s sit in an L. That region is complete, so no neighbouring cell can be a 3.
A region short of its number has to grow through empty cells. Flood outwards from it and count how many cells it could possibly claim; if that total falls short of its number, whatever boxed it in is wrong.
Example: A lone 5 sits in a pocket with only three empty cells around it. Something you placed earlier has sealed it off — go back and open the pocket.
The other half of the same count: if a region can reach exactly as many cells as its number and no more, every one of those cells belongs to it. There is no choice left to make.
Example: A 4 with one cell placed can only reach three further cells before hitting finished regions. All three are 4s.
Two equal clues near each other are either one region or two, and the difference is decisive. If joining them would make a block bigger than the number allows, they must stay apart — which means the cells between them belong to something else.
Example: Two 3s sit two cells apart on a row. Joining them through the gap would make four cells in a 3-region, so the cell between them is not a 3.
Only shared sides join cells into a region. Two equal numbers meeting diagonally are two separate regions and are perfectly legal — do not rule out a placement just because it puts matching numbers corner to corner.
Example: A 2-region ends at one cell and another 2-region starts on the diagonal below it. Both are fine as long as neither pair shares a side.
A cell in a corner has only two neighbours, and one along an edge only three. That limits both how a region can grow into it and which regions can reach it at all, so the border of the board is usually where the next certain move is.
Example: The corner cell touches two cells, both already in finished regions of 3 and 5. The corner is a region of its own, so it is a 1.
A region may hold no clue at all, or several. Do not assume each clue belongs to its own region, and do not assume an unclued patch is a mistake — some regions are meant to be deduced entirely from the space left over.
Example: A pocket of four cells is surrounded by finished regions and holds no number. It has to be a region of 4.