[iagslrum] walk through a wall, once

given a maze of walls, let the player be permitted to walk through a wall exactly once.  what is the shortest solution?  probably a simple application of breadth first search.  what pair of locations maximize the minimum path?  all-pairs shortest path will find this, but is there anything better like there is for trees?

previously, walking through as many walls as you want.

what if you pick the wrong wall to walk through?  maze could (easily) be designed to be solvable without walking through any wall -- walking through a wall just makes things faster.  or (this requires thought), maze is unsolvable without walking through a wall, but walking through any wall works: two intertwined connected components.