Abstract
This problem originates from Chemistry. You will soon find that the Oxygen and Hydrogen in the image can be replaced with vertices and arrows, which is why I propose it here. Although its main purpose is to practice Chemistry, the Mathematics part of it is absolutely not trivial.
Problem
Given a net of tetrahedra and their centers. The edges of the tetrahedra themselves are not matched, but the lines connecting the vertices and the centers of the tetrahedra. To illustrate, let's have a look at the example given in the book:
Ignoring the black points, mark each edge with an arrow so that, for each vertex, there are 2 arrows coming from and 2 arrows coming to it. Define a box the outer boundary of the net, so any edge that is not inside or lies on the surface of the box is not counted. How many ways are there to mark the edges of a $n \times n \times n$ box with arrows? (The illustration above is a $3 \times 3 \times 2$ box)
I attempted to convert the problem to a directed Graph on the plane, with the degree of each edge equal to 4, and each pair of edge from a vertex is also a pair of exactly one hexagon. But I feel that the conditions are not tight enough to solve this problem.
Any idea would be appreciated.
Edit: I have indeed found an estimated answer to the question here (please scroll to page 2 to read the mathematical solution), still I need a better explanation to the problem so that we can calculate the exact value of the answer.