Question

You want to paint several identically-sized cubes so that each face of each cube is painted with exactly one color and each cube is painted with six different colors. how many different cubes can you produce if you have

Answer 1

To determine the number of different cubes that can be painted with six unique colors, one per face, calculate 6 factorial and divide it by the number of rotations that result in identical cubes, resulting in 30 unique cubes.

Step-by-step explanation:

The question asks about the number of different ways you can paint identically-sized cubes with six different colors, with each face of the cube painted a distinct color. This is a combinatorics problem that can be solved using mathematics, specifically factorials and permutations.

Consider the first cube face; it can be painted in 6 different colors. After the first face is painted, there are 5 remaining colors for the second face, 4 for the third face, and so on until there's only 1 color left for the last face. The total number of ways to do this for one cube would be 6 x 5 x 4 x 3 x 2 x 1, which equals 720.

However, since cubes can be rotated, painting different faces first does not necessarily result in a unique cube. Any cube can be rotated to place any face on the bottom. Once the bottom face is fixed, there are 4! (4 x 3 x 2 x 1 = 24) ways to arrange the remaining colors (ignoring the top), which are equivalent due to rotation around the vertical axis. So, the total number of unique cubes is the initial 720 divided by 24, yielding 30 unique cubes.