A cube has $12$ edges. Cut seven of them and lay out the remaining ones on a table. It's known that the number of distinct connected meshes with non-overlapping faces is $11$ (How many distinct ways to flatten a cube?).
The number of ways you can choose seven of the edges to cut is ${12 \choose 7}=792$. But a lot of them aren't valid spanning trees since some faces of the cube will be isolated from the rest.
I wrote some code to loop through the $792$ combinations and count the instances where the spanning tree property is preserved (and hence a valid connected mesh is possible).
This turned out to be $384$. This number is highly composite. It turns out that $384=4! \times 2^4$. This can't be just a coincidence. There must be a reason why it turns out to be such a nice number. I'm looking for the connection I can't see.
Perhaps something to do with the cube having $4$ main diagonals?
