Unique binary patterns on a Cube

Question

Given a cube where each face can be individually colored either black or white, I know there should be $2^6$ ways to color the cube.

How can I tell which cubes are duplicates of another cube but with a different orientation?

How can I tell how many unique cubes are there?

Answer 1

This is not so bad to count directly.

• There is only one all-white cube.
• There is only one cube that has a single black square (up to orientation).
• There are only two cubes that have two black squares (up to orientation). Either the black squares are on opposite faces or they are adjacent.
• The "hard" case: how many cubes have three black squares? Either they meet at a corner or two of the three are on opposite faces. So there are only two.
• Can you count how many have four black squares?
• Have five?
• Have six?