Fillomino rules
Fill every empty cell with a number so that each region of matching, connected numbers contains exactly that many cells.
The grid
A Fillomino puzzle is a square grid. Some cells start with a number in them; these are the clues, and they can't be changed. All other cells are empty. When the puzzle is solved, every cell holds a number, and the grid is divided into regions.
A region is a group of cells that hold the same number and are connected to each other through shared sides. Cells that only touch at a corner are not connected. So a region is a polyomino: a domino, a tromino, a tetromino, and so on.
The three rules
- Every region's number equals its size. A region of 3s has exactly three cells. A region of 5s has exactly five. A 1 is a region on its own.
- Regions of the same size can't share a side. If two groups of 4s touched along an edge, they would really be one group of eight 4s, which breaks rule 1. Regions of the same size may touch diagonally, corner to corner, because diagonal cells aren't connected.
- Every cell belongs to exactly one region. There are no blank cells in a finished grid.
Two things often surprise new players. First, a region doesn't need a clue in it. Some regions are completely empty at the start, and you work out both their shape and their number. Second, the same number can appear in many different regions, as long as those regions never share a side.
A small example
Here is a 4×4 puzzle with six clues, and its solution. Thick lines in the solution show where one region ends and the next begins.
Walk through the solution and check each rule. The 3 in the top-left corner grows right into a row of three 3s. The 2 below it has only one way to grow, downward, because the 3 and the 1 block the other sides. The 4 becomes a 2×2 square. The three 1-clues stay alone.
Two regions in the solution had no clue at all: an L-shaped region of 3s along the bottom, and a 1 in the bottom-left corner. The two regions of 3s are both legal because they never share a side. Unclued regions like these are normal in Fillomino, and you can always prove what they must be. Once the regions around them are finished, the leftover cells can only be divided in one way. The Enclosed areas article shows how.
How solving works in the app
In the app you tap a number in the toolbar to arm it, then tap or drag across empty cells to fill them. When a region is complete and correct, it turns green. The check button highlights any cell whose region is the wrong size, and undo steps back one move at a time.