(2nd round) Alper is the owner of Cube-XYZ, a perfect cube-shaped planet, and builds a mansion on one vertex of the cube, and distinct houses on the seven other vertices. houses are considered next door if they are connected by an edge of the cube. He is to host an intergalactic tea party for his friends: Wes, Ethan, Henry, Alan, James, Vincent, and Tom. They are able to be accommodated in separate houses. As host, Alper has conditions on how the guests must be placed on his planet: i) His best friend Ethan must live next door to him. ii) Henry and James must not live next door to each other. iii) Alan and Vincent must live next door to each other iv) Wes should be as many edges away from Alper as possible v) Alper lives in the Mansion vi) Tom lives (strictly) closer in distance to Wes than Alper. i. Given all conditions are met, how many combinations of living arrangements are there? Possible solutions are: 24, 36, 45, 48, 64.
