# Polyominoes

### Ominoes

Dominoes is a game played with playing pieces made from two adjoining squares.

Tetris is a computer game played with playing pieces made from four adjoining squares.

Pentominoes is a game played with playing pieces made from five adjoining squares. There are 12 "free" pentominoes – ones which can be turned over to make their own mirror image.

There are lots of different polyominoes – 36 hexominoes (six squares), 108 septominoes (made from seven squares), etc, with the number of pieces increasing as the number of squares used to make them increases.

 Number of squares Name Number of free polyominoes Number of fixed polyominoes 1 monomino 1 1 2 domino 1 1 3 tromino (or triomino) 2 2 4 tetromino 5 7 5 pentomino 12 18

Whereas dominoes is normally played using numbers on the individual squares, pentominoes uses the shapes themselves, and can be either a puzzle or a board game.

### Puzzle

Try to fit the pentominoes into a rectangular area. The easiest is 6 x 10 (the closest rectangle to a square that can be made with the pieces), the hardest 3 x 20.

 Area Number of Solutions 6 x 10 2,339 5 x 12 1,010 4 x 15 368 3 x 20 2

### Board Game

Two players take turns placing pentomino pieces on an 8 x 8 board (eg, a chess board), with the object being to block your opponent from being able to place a piece.

FWIW the 12 pentomino pieces can be all be placed on an 8 x 8 board leaving four squares uncovered.

### Extension

• There are 11 hexominoes which can be folded up to make cube. Which are they?
Answer: See this graphic.

• A complete set of 35 hexominoes has a total of 210 squares, but it is not possible to pack them into a rectangle. How can we know this?
Hint: What colours do individual hexominoes cover if laid on a chess board? Do you want an odd number of each colour or an even number?
Answer: (Highlight to read.) If the hexominoes are placed on a checkerboard pattern, then 11 of the hexominoes will cover an even number of black squares (either 2 white and 4 black or vice-versa) and 24 of the hexominoes will cover an odd number of black squares (3 white and 3 black). Overall, an even number of black squares will be covered in any arrangement. However, any rectangle of 210 squares will have 105 black squares and 105 white squares.

• A complete set of 108 heptominoes has a total of 756 squares but it is not possible to pack them into a rectangle. How is it easy to prove this?
Hint: Look at the shape of one particular heptomino.
Answer: (Highlight to read.) One heptomino makes it impossible, since it has a hole in the middle which cannot be filled by another heptomino.

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