8. Each face of a cube has a different color (the six colors are fixed), how many different coloring methods are there?


8. Let the six colors be called $1,2,3,4,5,6$. Place the cube on the table with the 1-colored face down. Consider the 2-colored face; if it is up, it can be rotated around the vertical axis to bring the 3-colored face to the front. This fixes the cube. The other faces can be colored in $3!=6$ ways. If the 2-colored face is adjacent to the 1-colored face, rotate the cube to bring the 2-colored face to the front, fixing the cube. The remaining faces can be colored in $4!=24$ ways. In total, there are $6+$ $24=30$ different coloring methods.