When you remove this three-dimensional diagram from the apex of the pyramid, you can "flatten" it into a plane figure with the original bottom in the middle and the original edges on four sides (like a four-petal flower). This deformation will not change the connectivity between faces, so it will not change the result of coloring.
This has become a common problem of map coloring in high school mathematics, which may be helpful for your intuitive understanding.
Back to the point. According to symmetry, the relationship between the bottom and the four sides is different, but the four sides are symmetrical to each other, so they can be considered equivalent.
Assuming that the color of the bottom surface has been selected, there are four selection methods.
Now draw the other four sides. Only the remaining three colors can be painted on all sides. This is a coloring problem of circular queue.
If the edges are different, let's assume that ABCD has four edges. There are three options for A, only two for B, three for C, only 1 for D (restricted by AC, D and B are the same color), and * * * is 3 * 2 * 3 =18; If all four sides are regarded as the same, and the same color is selected twice, there are three kinds. The remaining two kinds are arranged clockwise and counterclockwise, and * * * is 3*2 = 6 kinds.
Comprehensive bottom selection method and side selection method:
If the edges are ordered, one * * has four * *18 = 72 kinds;
If the two sides are not in order, that is, they are regarded as the same, there are 4 * * * = 24 kinds.
Note: I am different from the blue tropical fish. He chooses the side first. But the result is the same)
Supplement:
If the topic does not require that all four colors must be used up, then choose a profile in two cases:
1) uses two colors. There are three ways to choose color: c (3,2) = 3; If each side is considered different, there are two ways to draw it (abab or baba).
* * * meter: there are 3*2 = 6 different sides, and 3 different sides are regarded as the same.
2) Use all three colors. Ditto.
At this time, the final answer is:
If the two sides are in sequence, 4 * (6+18) = 96;
If there is no order on both sides, 4*(3+6) = 36 kinds.