firstname.lastname@example.org (Michael Michalchik) wrote in message news:<email@example.com>... > Kevin Saff <firstname.lastname@example.org> wrote in message news:<I808DL.GF2@news.boeing.com>... > > Michael Michalchik wrote: > > > I was wondering if anyone knows if all possible topologies of finite > > > 2d mazes can be solved by a finite algorithm. For example, we know > > > that all fully connected mazes can be solved by picking a wall and > > > exhaustively following it. Can a general solution work for all mazes > > > including the ones that are piecewise disconnected? If this is > > > possible, is the general solution a solved problem? > A maze with n walls is homeomorphic to the n-times punctured plane. So the method of cuts used by Cauchy to derive a simply-connected domain can be applied. The cuts in this instance become barriers joining the walls in some sequence, the last one getting a final barrier going off to infinity. Then any two points in the maze can be joined by a path which is homotopically unique. The method of Tremaux noted in the Wikipedia Maze article may amount to the same thing but with the barriers introduced on the run rather than in advance.