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?

It might be possible for N to be higher, but
N can be 25.

The cube can be oriented with the top number being the lowest, second lowest, middle, second highest or highest integer of the visible numbers. Of the remaining four visible sides, the lowest, second lowest, second highest or highest integer can be oriented toward the door. This gives 5*4 = 20 possible combinations.  A  different value of 1 to 20 can be assigned, with the assigned number indicating the sequence number of distinct integers of the list of N distinct integers (excluding the 5 integers visible on the cube). The assigned number would then represent the unseen number.

Posted by Dej Mar on 2013-07-11 08:21:19