This is a thought that I was having while building a model of a level 4 Menger sponge in minecraft.
Imagine a Menger sponge to be built of cubic voxels the same size as the smallest void. You can build a graph of this where each node is a voxel and there is an arc between them if that voxel is adjacent.
If you take this adjacency graph is there a hamiltonian path with in it?
For the level 1 case by simple inspection I've worked out no such path exists but I am unsure when we move to higher levels.