There is a calculation of the number of surface paths (with no backtracking allowed) between opposite corners of a Rubik's cube. I am interested in paths on an $L\times L\times L$ cube, where $L$ is the number of edges along the $\hat{x}$, $\hat{y}$ and $\hat{z}$ directions. There are two generalizations of the Rubik's cube problem that I am interested in. I am not hoping for an exact calculation but if there exist attempts or references for this, I would like to look at it.
- The first generalization is allowing paths through both the surface and bulk of the cubic lattice. So the more general question would be: how many paths exist between the opposite corners of the $L\times L\times L$ cube?
- The second generalization is to consider paths between opposite boundaries and not just opposite corners of the cube. In other words, can the calculation be generalized to the number of paths between two opposite boundaries of a cube? More generally, one could consider a cuboid with dimensions $L_x\times L_y \times L_z$
Are there analytical expressions known for these number of paths? Is there a reference?
Some clarifications based on discussion in the comments: you are allowed to travel every edge only once but you can revisit the nodes.