2. Color the six vertices of a regular pentagonal pyramid using at most six different colors, such that the two endpoints of any edge are colored differently. If one coloring method can be obtained from another by rotation, they are considered the same. How many different coloring methods are there?
(Romania)
2. Since the problem does not specify the number of colors to choose from, we will calculate the coloring methods based on the actual number of colors used.

If fewer than four colors are used, the bottom face can have at most two colors. Since 5 is an odd number, the colors of the bottom face vertices cannot be alternated. In this case, there are no valid coloring methods.

If six colors are actually used, without considering rotational equivalence, there are $6!=720$ methods. After removing rotations, there are 720 $\div 5=144$ methods.

If five colors are actually used, the coloring of the bottom face vertices must be in the form $A B A C D$. By rotation, the color $B$ can be moved to a fixed vertex, so the number of coloring methods is the permutation of 5 colors, which is $5!=120$.

If four colors are actually used, the coloring of the bottom face vertices must be in the form $A B A B C$. By rotation, the color $C$ can be moved to a fixed vertex, so the number of coloring methods is the permutation of 4 colors, which is $4!=24$.
In summary, the total number of coloring methods is
$$
144+120+24=288 .
$$