You are and your friend 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, its 6 sides parallel to the walls of the room. Your friend 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 6 sides kept parallel to the walls of the room. Your friend will then be sendt away, after which you can enter the room and are allowed to observe the 5 visible sides of the cube.
What is the largest N that guarantees you to be able to determine the number on the bottom of the cube and what should you instruct your friend to do with the cube for that N?
N=29 is possible see! still in need of strategy http://mathoverflow.net/questions/52541/hard-cube-puzzle