MATHCOUNTS Exercises – 15

Solution First, there is only one way to paint the cube with all opposite faces painted in the same color, as shown below, with the front face colored in red:

Second, there are 3 ways to paint the cube with one pair of opposite faces in the same color, while the other two pairs of opposite faces in different colors, as shown below, with the front face in the same color of the back face:

Finally, there are 2 ways to paint the cube without any pair of opposite faces in the same color, as shown below, with the front face colored in red:

Therefore, the final answer to the question is 6.