Fillomino

Fillomino is a grid puzzle where the player fills every empty cell with a number so that each group of connected identical numbers forms a polyomino (a connected shape) whose size matches the number. For example, all the 3s that touch each other must form a group of exactly three cells. Numbers are given as clues in some cells, and the solver must complete the grid so that every cell belongs to a correctly-sized group. Two groups with the same number may exist in the grid, but they must not be adjacent horizontally or vertically.

Rules

1. Fill every empty cell in the grid with a positive integer.
2. Each group of connected cells (horizontally/vertically adjacent) containing the same number must have exactly that many cells. For instance, a group of 4s must contain exactly four cells.
3. Two distinct groups of the same number may not be adjacent horizontally or vertically (they would merge into one oversized group).
4. There is no upper limit on the numbers that can appear; groups of 1, 2, 3, or larger sizes are all valid.
5. Some cells contain pre-filled clue numbers.

Strategies

• Clue Expansion: A clue number tells you the exact size of its group. Start by considering how the group could extend from that cell.
• Ones Are Instant: A cell with the number 1 forms a complete group by itself. All of its neighbors must contain different numbers.
• Twos Need One Neighbor: A cell with a 2 must connect to exactly one other cell containing a 2. If it has only one possible direction to expand, that placement is forced.
• Separation Rule: Two clue cells with the same number that are adjacent must belong to the same group. If joining them would create a group larger than the number, there is a contradiction — reconsider earlier placements.
• Barrier Detection: If a group has reached its target size, all cells adjacent to the completed group that do not belong to it cannot contain that number.
• Small Spaces: If a confined area of the grid can only hold a certain number of cells, the group size in that area is constrained.

Fillomino was created by the Japanese puzzle publisher Nikoli, which introduced it in their puzzle magazines in the late 1990s. The puzzle is sometimes also referred to as "Allied Occupation" in English, though the name "Fillomino" (a play on "fill" and "polyomino") has become standard internationally.

Nikoli is renowned for developing and refining logic puzzles with simple rules and deep logical complexity, and Fillomino is a prime example of this philosophy. The puzzle stands out because it does not restrict which numbers can appear — unlike Sudoku where digits 1-9 are used, Fillomino allows any positive integer, and the grid can contain multiple groups of the same size as long as they do not touch.

Fillomino has been a regular feature in the World Puzzle Championship since its inclusion in the competition roster. It is prized by competitive solvers for its flexibility and the way it rewards spatial reasoning. The puzzle also appears frequently in online puzzle communities such as Logic Masters and Puzzle GP events.

The puzzle has an interesting mathematical property: because groups of the same number cannot be adjacent, the grid effectively partitions into polyominoes whose shapes are determined entirely by logic. This connection to polyomino theory has attracted interest from recreational mathematicians. Variant forms include "No Rectangles Fillomino" (groups cannot form rectangles) and "Fillomino with walls" (some cell borders are pre-drawn).