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The planet Cube is an ordinary cube with 6 faces, pilgrims must visit 6 towns, one on each face. The length of any edge of Cube is 3 miles, the cities have the following coordinates (x,y,z) in miles:
A (2,0,2)
B (3,1,1)
C (1,3,2)
D (0,2,1)
E (1,1,0)
F (1,1,3)

Pilgrims have to follow the shortest closed path on the surface of planet Cube going from A to A traversing each town: your task is to find the sequence of visited towns and the total distance traveled by pilgrims in miles, describing the details.

Here is the image of planet Cube using transparency to show all towns (towns B,C and E are not directly visible from the observer):

image of Cube and cities

Town A is located on front face, town F on top face, town E on bottom face, town D on the left, town B on the right and town C on the back. Please refer to the coordinates. Note that there are no tunnels, i.e. pilgrims walk on the surface of each cube face.

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This turns out to be fairly straightforward.

First we need to know the distance between any pair of cities.

You need to unfold the cube to find the surface distance between two cities. For example, for A-B, unfold the front and right faces to find that the straight line path between them goes 2 steps in one direction (actually steps x+1 and y+1) and 1 step in another (step z-1). By Pythagoras, that makes their distance $\sqrt{2^2+1^2}=\sqrt{5}$.
For cities on adjacent faces, I have not found any cases where it is shorter to cut across an intermediate face. For example, B-F is 4 steps directly (z+2, x-2), while cutting across the front face would make it $\sqrt{3^2+3^2}=\sqrt{18}>4$.
In the table below I have the squares of the distances I found. 

   A B  C  D  E  F
 -+-+-+--+--+--+--+
 A|-|5|26|17|10| 5|
 B| |-|17|26| 9|16|
 C| | | -| 5|16| 9|
 D| | |  | -| 5|10|
 E| | |  |  | -|25|

Now we need to find the shortest path.

This turns out to be trivial. The six shortest edges have lengths $\sqrt{5}$ and $\sqrt{9}$ (four of the former, two of the latter). These six edges form a cycle ABEDCFA or in reverse AFCDEBA. This must be optimal because there are no shorter edges.
The total length is $6+4\sqrt{5}$.

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  • $\begingroup$ Perfect answer! $\endgroup$ Commented May 7 at 15:44
  • $\begingroup$ Unfolding the cube is not a straight forward observation if you've never seen it before! $\endgroup$ Commented May 9 at 13:32
  • $\begingroup$ $\sqrt{9}$, also known as 3. :-) But you are right, $\sqrt{9}$ makes it clearer. $\endgroup$ Commented May 10 at 0:27

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