Fillomino
Divide the grid into polyomino blocks where each cell's number equals the area of its block.
Basic Game Premise & Rules
MechanicsFillomino is an untimed, deterministic pencil-and-paper logic puzzle played on a grid. Every valid puzzle has exactly one unique solution that can be discovered through rigorous deductive reasoning alone, without trial and error or guessing.
- The grid contains some initial numerical clues in certain cells; other cells are empty.
- Fill every empty cell with a positive integer such that the entire board is tiled into connected polyomino blocks.
- Each block of size K must contain the number K in every one of its constituent cells (e.g., three 3s form a 3-cell block; five 5s form a 5-cell block).
- Non-Merging Constraint: Two separate polyomino blocks of the same size cannot share an orthogonal edge (otherwise they would merge into a larger block). They may, however, touch diagonally at a corner.
- Clues do not have a 1-to-1 correspondence with blocks: a single block may contain multiple clues of the same number, or even no given clues at all (an emergent 'hidden' polyomino).
Rules by Example
See how each rule looks when done correctly versus when violated:
Block Size Matches Its Number
Connected cells with the same number form a block. Every block must contain exactly as many cells as the number inside it.
Same-Numbered Blocks Cannot Share an Edge
Two separate blocks with the same number cannot touch along a side (they would merge into an oversized block). They may only touch diagonally at corners.
History & Origins
Origins & EvolutionFillomino was created by Japanese puzzle creator 'Miku' and first appeared in *Puzzle Communication Nikoli* #47 in Autumn 1994. The name is a blend of the English word *fill* and the mathematical term *polyomino*.
Unlike traditional regional division puzzles (such as Shikaku) where every region must contain exactly one clue, Fillomino's genius lies in its emergent geometry: blocks can form around multiple given numbers or coalesce in empty spaces without any initial clue whatsoever. This dynamic made it an immediate international sensation. When introduced to North America as **Polyominous** in *Games Magazine*, it became one of the most frequently solved pen-and-paper puzzles, and it remains a staple event in the annual World Puzzle Championship (WPC).
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Fillomino formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Polyomino Decomposition & Percolation
Fillomino studies the partitioning of a planar grid graph G = (V, E) into vertex-induced connected components V_1, V_2, ..., V_m such that each component V_i satisfies |V_i| = k_i, and all vertices v ∈ V_i have label k_i. The non-merging rule dictates that if V_i and V_j share an edge in G, then k_i != k_j. This connects directly to percolation theory and cluster growth on 2D square lattices.
NP-Completeness via Planar 3-SAT
Deciding whether a Fillomino grid has a valid completion was proved to be NP-complete by Laura McPhail (2007). The proof constructs complex logic gates (wires, splitters, NOT gates, and clause checkers) out of interacting polyominoes and reductions from Planar 3-SAT. Takahiro Seta and Takayuki Yato (2002) also proved that the Another Solution Problem (ASP) for Fillomino is ASP-complete, showing that finding alternative solutions is computationally intractable.
Dual Boundary Planar Graphs
Between distinct polyominoes lies a set of boundary line segments forming an embedded planar graph on the dual grid. Every internal junction of four grid cells has degree constraints: the number of incident boundary segments cannot equal 1, because a boundary line cannot terminate inside the grid without creating an open polyomino.
Area Starvation & Expansion Heuristics
Human and algorithmic solving relies on space conservation: a nascent block of size K with only c < K cells must have at least one open boundary cell through which it can expand. If all but one exit is blocked by different numbers or walls, that remaining exit is forced to become part of the block ('escape path'). Conversely, surrounding empty cells must avoid exceeding the target area.
References & Further Reading
Bibliography- McPhail, L. (2007). 'Fillomino is NP-complete'. Technical Report, Department of Computer Science.
- Yato, T., & Seta, T. (2002). 'Complexity and Completeness of Finding Another Solution and Its Application to Puzzles'. IEICE Transactions on Fundamentals, E86-A(5), 1052–1060.
- Nikoli (1994). Puzzle Communication Nikoli #47. Tokyo: Nikoli Publishing.
- Friedman, E. (2009). 'Pencil Puzzles: A Survey of NP-Completeness'. Stetson University Mathematics.
- Demaine, E. D., & Hearn, R. A. (2008). 'Playing Games with Algorithms: Algorithmic Combinatorial Game Theory'. arXiv:0806.0778.
Free Printable PDF Booklet
Download our beautifully formatted, high-resolution A4 puzzle booklet. Generated directly from the handcrafted levels on AZTest, it contains 50 puzzles ordered from gentle introductory boards up to master-level challenges, followed by complete solutions at the back.
| Difficulty Stage | Grid Size | Description | Puzzles |
|---|---|---|---|
| Intro | 5×5 | Introductory grids teaching block expansion and non-merging rules. | 3 |
| Easy | 5×5 to 6×6 | Small polyominoes (dominoes, trominoes, tetrominoes) with clear escape routes. | 12 |
| Medium | 7×7 to 8×8 | Emergent blocks and dead-end avoidance. | 12 |
| Hard | 8×8 to 9×9 | Large blocks (6s, 7s, 8s) requiring multi-branch lookahead. | 12 |
| Expert | 9×9 to 10×10 | Master polyomino puzzles with extensive blank expanses. | 11 |
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