Lesson A bug lives on a corner of a cube

A bug lives on a corner of a cube

Problem 1

A unified geometric-combinatorial description can be created for all possible configurations.


        


The cube has 6 faces that can be combined in opposite pairs (front, back), (left, right), and (upper, lower)
in parallel planes.

Each path under the imposed conditions comprises an open loop of three sequential edges (four vertices) 
on one face of the cube and a similar open loop of three sequential edges (four vertices) on the opposite face, 
traversed in mutually opposite directions. 
These open loops on opposite faces are connected by connecting edges — or "bridges" — which are 
also traversed in mutually opposite directions.

Thus, for example, in the cube 12345678 in the figure, the first open loop of three edges (four vertices) 
is the path 1234, while the second open loop of three edges (four vertices) is the path 5678.
The first bridge is 45; the second bridge is 81 - it closes the whole path from 1 to 1.


OK, very good.


Now, there are 3 edges emanating from vertex 1: 12, 14, and 18.

For each of these edges, there are two open loops starting from these edges:

   1278 and 1234 for edge 12;
   1432 and 1458 for edge 14;  
   1872 and 1854  for edge 18.


Thus, we have 2*3 = 6 open loops listed above, created by edges 12, 14, and 18,
when they are starting edges in these open loops.
It creates/generates 6 possible paths satisfying the imposed conditions.


The other 6 paths arise when edges 12, 14, and 18 are the bridges for the other open loops.


In all, it gives 6 + 6 = 12 possible paths satisfying the imposed conditions.


ANSWER.  There are 12 possible paths for the ant satisfying the imposed conditions.