In class, we learned how, when order matters, we can count different possible objects by determining all the equivalent representations based on allowed transformations of one example. Here are some practice problems.

- In class, we showed that in flatland, there are two different ways to color a triangle's corners. Now, let's do squares.
- How many representations are there for a square with 4 differently coloredcorners?
- How many different transformations of a given square are allowed in flatland?
- How many different square colorings are there in flatland? List them all.

- Now, what about other regular polygons in flatland?
- How many ways are there to color a pentagon's 5 corners in flatland?
- How many ways are there to color an n-gon's corners in flatland? Find a formula.

- In sphereland, were we live, we can, of course, move things in 3 dimensions.
- How many ways are there to color a square's 4 corners in sphereland?
- How many ways are there to color a pentagon's 5 corners in sphereland? List them all.
- How many ways are there to color an n-gon's corners in sphereland?
- How many ways are there to color the corners of a rectangle in sphereland?
- A brick is a solid with 6 faces and 8 vertices, all faces at right-angles to each other, but the length, width, and heighth all different. How many different ways are there to color the corners of a brick in sphereland?
- Suppose that instead of the corners, we color the faces of the brick. How many representations are there for a colored brick in sphereland? How many different face-colored bricks are there in sphereland? Is the number different than what we get coloring corners? Explain.

- Some things that we count as different in 3 dimensions are actually the same in the 4-dimensional world.
- How many ways are there to color a 3-d brick's faces in 4 dimensions? (Imagine the brick is actually hollow and sewn from fabric so you can turn it inside out.)