Polyominoes

Polyominoes are geometric shapes formed by joining unit squares edge to edge. The term encompasses all such shapes regardless of size: monominoes (1 square), dominoes (2), trominoes (3), tetrominoes (4), pentominoes (5), hexominoes (6), and beyond. In a polyomino puzzle, the player must fit a given set of irregularly shaped pieces into a target shape — a rectangle, a frame, or any other outline — covering every cell exactly once. The generic nature of polyomino puzzles allows for infinite variety in piece sets and target shapes.

Rules

1. You are given a target shape (grid outline) and a set of polyomino pieces.
2. Place all pieces into the target shape so every cell is covered.
3. Pieces may typically be rotated and flipped freely (rules vary by puzzle).
4. Pieces must not overlap.
5. No cell of the target shape may be left empty.

Piece Categories by Size

| Name | Squares | Count (free) |

|------|---------|-------------|

| Monomino | 1 | 1 |

| Domino | 2 | 1 |

| Tromino | 3 | 2 |

| Tetromino | 4 | 5 |

| Pentomino | 5 | 12 |

| Hexomino | 6 | 35 |

| Heptomino | 7 | 108 |

| Octomino | 8 | 369 |

Strategies

• Area Matching: Calculate the total area of all pieces and confirm it equals the area of the target shape. If not, the puzzle is unsolvable or you have the wrong piece set.
• Corner Filling: Corners of the target shape are the most constrained locations. Identify which pieces can fit into corners and place those first.
• Parity Coloring: Color the grid like a checkerboard. Each piece covers a specific ratio of black to white squares. Verify that the total black/white counts of all pieces match the target.
• Constrained Pieces First: Pieces with unusual shapes (crosses, U-shapes, asymmetric shapes) have fewer valid placements. Position them before more flexible pieces.
• Avoid Orphan Regions: As you place pieces, ensure no isolated region is created that is smaller than your smallest remaining piece.
• Boundary Tracing: Work along the boundary of the target shape, filling inward. This naturally constrains the problem and prevents large ambiguous interior spaces.

The mathematical study of polyominoes was pioneered by Solomon W. Golomb, who introduced the term in a 1953 talk to the Harvard Mathematics Club and published a formal paper on the subject the following year. His 1965 book "Polyominoes: Puzzles, Patterns, Problems, and Packings" became the definitive reference and sparked decades of mathematical research.

However, the concept of fitting geometric shapes together predates Golomb by centuries. Domino puzzles date back to the 18th century, and tiling problems have been studied in mathematics since antiquity. The specific study of square-based shapes gained momentum in the 20th century alongside the development of combinatorics and recreational mathematics.

Martin Gardner popularized polyominoes through his "Mathematical Games" column in Scientific American beginning in 1957. His accessible writing brought these shapes to a wide audience and inspired generations of mathematicians, puzzle designers, and game creators. The influence of polyominoes on popular culture was cemented when Alexey Pajitnov used tetrominoes as the basis for Tetris in 1984.

Polyomino tiling problems have become fundamental objects of study in computer science and mathematics. Determining whether a given set of polyominoes can tile a given region is, in general, NP-complete — making it one of the classic hard problems in computational complexity theory. Research continues into tiling theory, enumeration of polyominoes, fault-free tilings, and connections to statistical mechanics. Modern polyomino puzzles appear in board games (like Blokus), video games, educational tools, and competitive mathematics events worldwide.