Take a sheet of graph paper (the bigger the squares, the better). What shapes can you cut out using the squares?
A domino is a shape made from two adjacent squares. These are dominoes:
By modifying the word domino, we can get names for similar shapes that use different numbers of squares. First, we have the lonely monomino, made of just a single square:
If we use three connected squares, we get a tromino (or a triomino). These are the trominoes:
Four connected squares make a tetromino. These are the tetrominoes:
There are names for larger shapes like this, too. Five connected squares make a pentomino, six connected squares make a hexomino, seven connected squares make a heptomino, and so on; these names come from the Greek words for five, six, seven, etc.
In general, a polyomino is any shape formed from connected squares like this. So dominoes, trominoes, and tetrominoes are just different sizes of polyominoes.
Above we listed one monomino, two dominoes, six trominoes, and 19 tetrominoes. Let’s call this way of distinguishing polyominoes “Convention A” (I just made this name up; this isn’t a technical math term). However, this is not the only way to think about polyominoes. There are two other common ways to distinguish polyominoes.
Usually we like to think of the two dominoes as the same shape, because we can get one of them from the other by just rotating it 90 degrees. Let’s call this “Convention B.” If we follow this convention, there is essentially just one domino, not two:
Likewise, we often consider the two “straight” trominoes to be the same, and the four “bent” trominoes to be the same, because we can get the others by rotations. If we think about things this way, we have essentially just two trominoes, not six:
And we have essentially just seven tetrominoes, not 19:
Notice that the last two pairs of tetrominoes above look very similar, but we cannot rotate one of them to get the other. Instead, we would have to pick up the shape and flip it over. So, according to Convention B, we consider them to be different shapes. (These seven shapes are the familiar Tetris pieces. If you’ve played Tetris, you are familiar with the idea of being able to rotate pieces but not flip them over.)
In the third common way of thinking about polyominoes, which we shall creatively call “Convention C,” we allow rotations and reflections (flipping pieces over). In other words, if one polyomino can be turned into another by rotating it, picking it up and flipping it over, or both, then we consider the two polyominoes to be the same. This brings the number of tetrominoes down to just five:
Sometimes polyominoes are called “fixed” when we think about them using Convention A, because they can be neither rotated nor flipped over. (If we rotate one or flip one over, we get a different polyomino, according to Convention A.) Polyominoes are called “one-sided” when we use Convention B, because they can be rotated but not flipped over, and they are called “free” when we use Convention C, because we are free to rotate them or flip them over however much we like.
Here are some questions to think about. I don’t expect you to be able to give me answers to all of these questions, because they take quite a bit of thought and experimentation, but hopefully you can answer at least one or two of them.
It will probably be helpful to get some graph paper to draw on. Cutting out polyominoes from graph paper is also helpful, so that you can play with them on a desk to help you visualize how they fit together.
The area of a polynomino is the number of squares it contains: the area of the monomino is 1, the area of a domino is 2, and so on. How many polyominoes are there of a given area?
To answer this question, we need to choose one of the conventions described above so that we can say when two polyominoes are different. Our results so far are summarized in the following table.
| Name | Area | Convention A | Convention B | Convention C |
|---|---|---|---|---|
| Monomino | 1 | 1 | 1 | 1 |
| Domino | 2 | 2 | 1 | 1 |
| Tromino | 3 | 6 | 2 | 2 |
| Tetromino | 4 | 19 | 7 | 5 |
| Pentomino | 5 | |||
| Hexomino | 6 |
Choose one of these three conventions. List all of the different pentominoes to figure out how many there are. (Be careful that you aren’t listing any shape twice and that you aren’t missing any shapes.) Can you find a pattern in these numbers? If so, make a prediction for how many hexominoes there are, according to the pattern you found. Then list all of the different hexominoes. Was your prediction correct?
For a real challenge, try to list all of the heptominoes (polyominoes with area 7). Be careful—one of the heptominoes has a hole in it!
We can also think about the perimeter of a polyomino, which is the distance around the outside edge. We consider each of the little squares to have a side length of 1. So, for example, the monomino has a perimeter of 4, and a domino has a perimeter of 6. How many polyominoes are there with a given perimeter?
Again, to answer this question, we first need to choose one of the conventions described above so that we can say when two polyominoes are different. It is easy to see that the smallest possible perimeter is 4, because that is the perimeter of a single square. So, if we make a table like the one above, we can start with 4:
| Perimeter | Number of polyominoes |
|---|---|
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 | |
| 9 | |
| 10 | |
| 11 | |
| 12 |
Start classifying the polyominoes by their perimeter and complete the table above. (Don’t forget to specify which convention you’re following.) Can you find any patterns in these numbers?
For a given area, what shape of polyomino gives the largest perimeter? What shape gives the smallest perimeter? If you like to think about things algebraically, let n represent the area of a polyomino. Can you find an algebraic expression in terms of n that gives the largest or smallest perimeter possible for a polyomino with area n?
Which polyominoes can tile the plane?
In other words, if you have a big box filled with a bunch of copies of one particular polyomino, can you fit them together to cover a large floor without any gaps or overlap? For some polyominoes, the answer is yes, but for other polyominoes the answer is no. Can you find some common characteristic shared by those polyominoes that can tile the plane?
It is really helpful to cut out some paper polyominoes to play with when you are trying to answer this question.
Choose one of the three conventions described above, and choose a particular size of polyomino (for example, the tetrominoes). Now suppose you have one copy of each of these polyominoes. Can you fit all of them together to make a rectangle without any gaps or overlap? (A good place to start is to figure out what the total area of such a rectangle would have to be, so that you can come up with a list of possible dimensions for the rectangle.)
What if you have two copies of each polyomino? Three copies?
Suppose that you have an 8 × 8 checkerboard and a bunch of dominoes (32 of them, to be precise), sized so that the squares that make up the domino are exactly the same size as the squares on the checkerboard. Can you find a way to tile the checkerboard with the dominoes? In other words, can you cover the checkerboard with the dominoes so that all of the squares are covered and no dominoes overlap or hang off the edge of the checkerboard?
Now suppose you remove one of the corner squares of the checkerboard, and then also remove the diagonally opposite corner square. You also throw away one of your dominoes. So you have 62 squares remaining on the checkerboard, and 31 dominoes. Now can you find a way to tile the checkerboard with the dominoes?
Go back to the original 8 × 8 checkerboard. You have a single monomino and a bunch of “bent” trominoes (the L-shaped ones). Choose any square of the checkerboard, and put the monomino there. Can you tile the rest of the checkerboard with the trominoes? Does it matter which square you put the monomino on?