# Equality and transformation groups

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.)