Polyominoes
ominoes  puzzle
 board game  extension
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 viceversa) 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.
