The Magic Cube Encryption

Question

I am trying to solve a very nice problem and I found a solution but this is more of like static solution.

The problem.

Cube has 8 corners, each containing one character. For example, the string “ABCDEFGH” would be represented on the cube like in the image below:

Cube can be rotated to the left, right, up and down.

This rightwards rotation converted the initial string “ABCDEFGH” into “BFGCAEHD”.

I want to know is there any algorithm or formula are there to find out changes in the corner? I did solve it by taking the notes of all changes of corners. For example, if A is in position 1 then if you change to right it will always replace it's position with E which is in position 5. So this is a very static solution. This solutions works because cube rotation is also static and never change position. However, want to know if this can be solved with a specific algorithm. Thank you :)

Answer 1

Not sure if I understand correctly, but I think you just need to write down the permutations for each case? Like, if you consider the three possible axes of rotation, and clockwise and counter-clockwise rotation directions, you have the six possible ways in which you can rotate the cube. For example, in Python you could do it like this:

def permute(s, perm):
    return ''.join(s[i] for i in perm)

def rotate_frontback_cw(cube):
    return permute(cube, [1, 5, 6, 2, 0, 4, 7, 3])

def rotate_frontback_ccw(cube):
    return permute(cube, [4, 0, 3, 7, 5, 1, 2, 6])

def rotate_leftright_cw(cube):
    return permute(cube, [4, 5, 1, 0, 7, 6, 2, 3])

def rotate_leftright_ccw(cube):
    return permute(cube, [3, 2, 6, 7, 0, 1, 5, 4])

def rotate_updown_cw(cube):
    return permute(cube, [1, 2, 3, 0, 5, 6, 7, 4])

def rotate_updown_ccw(cube):
    return permute(cube, [3, 0, 1, 2, 7, 4, 5, 6])

def cube2str(cube):
    a, b, c, d, e, f, g, h = cube
    return (f'   {h}--------{g}\n'
            '  /|       /|\n'
            ' / |      / |\n'
            f'{e}--------{f}  |\n'
            f'|  {d}-----|--{c}\n'
            f'| /      | /\n'
            f'|/       |/\n'
            f'{a}--------{b}')

cube = 'ABCDEFGH'
print('cube')
print(cube2str(cube))
print('rotate_frontback_cw')
print(cube2str(rotate_frontback_cw(cube)))
print('rotate_frontback_ccw')
print(cube2str(rotate_frontback_ccw(cube)))
print('rotate_leftright_cw')
print(cube2str(rotate_leftright_cw(cube)))
print('rotate_leftright_ccw')
print(cube2str(rotate_leftright_ccw(cube)))
print('rotate_updown_cw')
print(cube2str(rotate_updown_cw(cube)))
print('rotate_updown_ccw')
print(cube2str(rotate_updown_ccw(cube)))

This would print:

cube
   H--------G
  /|       /|
 / |      / |
E--------F  |
|  D-----|--C
| /      | /
|/       |/
A--------B
rotate_frontback_cw
   D--------H
  /|       /|
 / |      / |
A--------E  |
|  C-----|--G
| /      | /
|/       |/
B--------F
rotate_frontback_ccw
   G--------C
  /|       /|
 / |      / |
F--------B  |
|  H-----|--D
| /      | /
|/       |/
E--------A
rotate_leftright_cw
   D--------C
  /|       /|
 / |      / |
H--------G  |
|  A-----|--B
| /      | /
|/       |/
E--------F
rotate_leftright_ccw
   E--------F
  /|       /|
 / |      / |
A--------B  |
|  H-----|--G
| /      | /
|/       |/
D--------C
rotate_updown_cw
   E--------H
  /|       /|
 / |      / |
F--------G  |
|  A-----|--D
| /      | /
|/       |/
B--------C
rotate_updown_ccw
   G--------F
  /|       /|
 / |      / |
H--------E  |
|  C-----|--B
| /      | /
|/       |/
D--------A