Take a cube and cut $7$ of its $12$ edges. If the remaining faces are still all connected, it is possible to flatten out the cube onto a table as a 2-d mesh. The number of distinct meshes that come out is $11$ (see: How many distinct ways to flatten a cube?). And the total number of ways to make the $7$ cuts so that a mesh is possible is $384$ (it happens to simply be the number of spanning trees of the graph where the faces of the cube are nodes of the graph and edges of the cube are edges of the graph).
Another argument could be: let's start with one of the $11$ meshes. It corresponds to one set of $7$ cuts of the edges. Now, a cube has $48$ symmetries. So, it should be the case that applying any of the $48$ symmetries to the cube should result in $48$ cuts that lead to this same mesh. Which further implies that each of the $11$ meshes should result in $48$ ways to cut the cube edges. And the number of valid cuts of the cube should be at least $11 \times 48 = 528$. But this is larger than the $384$ we know to be true. It is also surprising that $384$ isn't even a multiple of $11$ (miss by $1$).
What am I missing here?
