The four bottom corners of a cube are colored red, green, blue, and purple.  How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.
1. **Color the bottom face**: There are \(4!\) ways to color the four bottom corners of the cube. This is because there are 4 colors and 4 corners, and each corner must be a different color. Therefore, the number of ways to color the bottom face is:
   \[
   4! = 24
   \]

2. **Determine the top face**: Once the bottom face is colored, the top face must be colored in such a way that each face of the cube has four different colored corners. Let's assume the bottom face is colored as follows:
   \[
   \begin{array}{cc}
   1 & 3 \\
   4 & 2 \\
   \end{array}
   \]
   Here, 1, 2, 3, and 4 represent the colors red, green, blue, and purple, respectively.

3. **Color the top face**: To ensure that each face of the cube has four different colored corners, the top face must be colored such that:
   - The color directly above 1 on the bottom face is different from 1, 3, 4, and 2.
   - The color directly above 3 on the bottom face is different from 1, 3, 4, and 2.
   - The color directly above 4 on the bottom face is different from 1, 3, 4, and 2.
   - The color directly above 2 on the bottom face is different from 1, 3, 4, and 2.

   Since the top face must also have four different colors, and there are only four colors available, the top face is uniquely determined by the bottom face. Specifically:
   - The color above 1 must be 2.
   - The color above 3 must be 4.
   - The color above 4 must be 3.
   - The color above 2 must be 1.

   This ensures that each vertical pair of corners (one from the bottom face and one from the top face) has different colors.

4. **Account for symmetries**: The cube can be rotated in space, and these rotations are symmetries that do not change the relative coloring of the corners. There are 24 such rotations (symmetries) of the cube. Therefore, we must divide the total number of ways to color the cube by the number of symmetries to avoid overcounting.

   Thus, the number of distinct ways to color the cube is:
   \[
   \frac{24}{24} = 1
   \]

Conclusion:
\[
\boxed{1}
\]