1. How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?
Answer: 8
Solution: If all vertices of the cube are of the same color, then there are 2 ways. Otherwise, look at a red vertex. Since it must have at least 2 red neighbors, there is a face of the cube containing 3 red vertices. The last vertex on this face must also be red. Similarly, all vertices on the opposite face must be blue. Thus, all vertices on one face of the cube are red while the others are blue. Since a cube has 6 faces, the answer is $2+6=8$.