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Hidden Number Guess (Posted on 2013-07-10) Difficulty: 4 of 5
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?

No Solution Yet Submitted by K Sengupta    
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Solution Functional Solution Comment 4 of 4 |
There are 24 ways to orient the cube. That plus five visible faces implies the maximum N is 29, which is attainable by adapting my soluton to 5 Cards Magic Trick Redux.

Alice calculates the sum of the faces mod 6.  This indicates which face should be hidden.  The largest face adjacent to the hidden face can be used to indicate which of the four possible values Bob should say.  One way is using N, W, S, E for 1st, 2nd, 3rd, 4th.  Bob then calculates his value mod 6 and notes which block to pick from.

A specific example: the faces of the cube are 2 opposite 9, 12 opposite 13, and 7 opposite 24.

Alice calculates (2+7+9+12+13+24) mod 6 = 1.  The second face, Face 7, is hidden.

The block list with 2, 9, 12, 13, 24 removed:
29 28 27 26 25 23 | 22 21 20 19 18 17 | 16 15 14 11 10 8 | 7 6 5 4 3 1

7 is in the last block and 13 is the largest face adjacent to it.  Then Alice orients Face 13 east.

Bob comes in and notes the block he needs is 7 6 5 4 3 1.

He calculates (2+9+12+13+24) mod 6 = 0, which implies the first number in the block, so Bob calls out the hidden Face is 7.

  Posted by Brian Smith on 2016-06-25 20:45:00
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