"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
A better generalization to 3D should assume a one-dimensional explorer, i.e. a two-dimensional path. Hold a piece of string in your hands, and drag it through a 3D maze from north to south while keeping one end of the string to the west of the maze and the other end to the east of the maze. (You may need a helper with tweezers to keep the string on the right path.) Again, either the maze has such a solution between two opposite faces, or the maze's dual has such a solution between two different opposite faces, never both and never neither.
That wasn't quite right. I apologize. Either there is a 2D path through the 3D maze, or there is a 1D path through the dual of the 3D maze (i.e. a support column), never both and never neither. Hence, symmetrically, there is either a 1D path through the 3D maze or there is a 2D path through the dual of the 3D maze (i.e. an airtight 2D wall). By "1D path" I mean one between two opposite faces, and by "2D path" I mean one through the other four faces. I now suspect that as 3D mazes become large without limit, there's always a 1D path between opposite faces unless p=0, and always a 1D path between opposite faces of the dual unless p=1.