## Hexpac online game

### Polyominos and Hexominos

Polyominos are simply connected sets of squares, the poly in the name refers to a variable number of squares making up the polyomino. Simply connected means that the square cells of the polyominos are connected to each other edge to edge as in a grid or chess board pattern. The types of polyominos (up to 7 cells) are defined by the number of cells in the polyomino. The mathematical law relating the polyominos type and the number in the set (i.e. the number of different varieties) is currently not known. The table below gives the numbers of polyominos up to the order 7.

Polyomino class | Number of cells | Number of different types |
---|---|---|

monomino | 1 | 1 |

domino | 2 | 1 |

tromino | 3 | 2 |

tetromino | 4 | 5 |

pentomino | 5 | 12 |

hexomino | 6 | 35 |

heptomino | 7 | 108 |

### Hexominos tiling the plane

The hexpac online puzzle game uses 35 hexominos, these comprise 24 odd hexominos and 11 even hexominos. Hexominos are either odd or even parity because if the hexominos are placed on a board with alternating white and black cells such as a chess board, the morphology of the hexominos will result in either an even or odd number of white and black chess board cells being covered. This property is useful for determining whether the set of the hexominos are theoretically able to completely tile a given area, for instance the 24 odd hexominos since even times odd results in an even value will be able to cover an even number of squares, the same argument applies to the odd hexominos, and also for the full set of 35 hexominos which will result in an even number of white and black cells. The 35 hexominos have 6 cells and total 210 cells. A rectangular area of this size would have to have 105 black and 105 white cells, an odd number. Therefore the complete set of 35 hexominos cannot be used to completely tile a rectangular area. A tiling of almost a complete rectangle is shown below together with a cross design. Patterns using the full hexominos set can not be made unless a chessboard colouring of the squares has a excess of squares of a single colour of the amount 2, 6, 10, 14, 18 or 22.

### Playing the Hexpac puzzle game

The Hexpac game consists of a 10 by 6 rectangle into which ten hexominos must be placed such that they completely tile the puzzle area. Some of the hexominos are already placed in the correct position to make the puzzle a bit easier and to give the puzzles variation. The hexominos can be moved by clicking and dragging the hexomino to the board. By dragging one of the edge cells the hexomino can be rotated. To place the hexomino on the board or to pick up the hexomino from the board click the hexomino. If instead of dragging the end cell it is instead clicked the hexomino will flip over, this is sometimes needed to fit a unsymmetrical hexomino piece on to the board. The puzzle is completed when the entire board is tiled with the hexomino shapes. There are a total of 49 puzzles to complete, the picture below shows all the available Hexpac puzzles.