Alex and Bob are given a list of N distinct integers and are told this:
Six distinct integers from the list are selected at random and placed one at each side of a cube. The cube is placed in the middle of a rectangular room in front of its only door, with one face touching the floor, 4 of its 6 sides parallel to the walls of the room.
Bob must enter the room and is allowed to alter the orientation of the cube, with the restriction that afterwards its in the same place with one face touching the floor and its 4 sides kept parallel to the 4 walls of the room. Bob will then be sent away, after which Alex can enter the room and is allowed to observe the 5 visible sides of the cube.
What is the largest N that guarantees that Alex will to be able to determine the number on the bottom of the cube and what should Alex instruct Bob to do with the cube for that N?