Also makes me wonder what a proof would look like.
Start with the simplest possible maze (i.e. one square with a start and a finish) which will be in two pieces. Then show that all possible additions will not add further pieces except for when you "cut" into an existing piece to make a separate non-solution route (i.e. a looped route that takes you back to the same point).
(Having any loops in the maze would break the condition of there being only one solution as you could take a detour round the loop as many times as you like and thus have an infinite number of "solutions")
The loop could be somewhere inaccessible from the starting point. Perhaps your logic holds if you insist that all pathways in the maze are connected, though.
The simple truth is that the original proposition "all mazes being two pieces" is just false.
Obviously false when you have multiple exists, or when there is only one exit/entrance and the goal is to reach the middle of the maze. But even if we stick to one entrance one exit mazes you can have multiple paths from exit to entrance which cut the maze into multiple pieces.
But even if we constrain ourselves to one entrance, one exit mazes with a single solution path you can trivially construct a maze inside a large room without connecting it to the walls. Thus creating a third piece to the maze.
It is just simply false.
I would be intrigued to see that "trivial" solution - how do you prevent there being two routes (and thus solutions) around the third piece without connecting it to the wall?
If you consider the same type of maze but have it in three dimensions, then all of them can be made out of just one piece, so it's clear that the definition of "maze" is too encompassing.
I think this is basically a flaw in the intuitive definition (or maybe only my intuitive definition) of "doubling back".
Now can we use THIS property to automatically solve a maze? make the WALLS of the MAZE themselves out of transparent TUBES and fill THEM with dyed water? An then the solved maze is simply the path that always has dyed walls on one side and transparent walls on the other. Not sure what the mathematical proof for this will look like but sounds nearly as good as the original "water can solve a maze" proof?
To use this property on a maze you'd have to traverse the whole maze using the typical cell adjacency, reconstruct it in a format that lets you bucket fill the walls, do that, and then also construct an algorithm which acts using the different colored walls. It only really works for a person interacting with a maze in a format where they can bring it into a picture editor.
This is true for mazes with only one path from start to end. Each additional path necessitates a separate piece (path permutations nonwithstanding)
One of my favourite bits of trivia: it is this property (being "unicursal") which leads to a maze being more specifically characterized as a labyrinth.
Wikipedia[1] also notes this contradiction:
> Although early Cretan coins occasionally exhibit branching (multicursal) patterns, the single-path (unicursal) seven-course "Classical" design without branching or dead ends became associated with the Labyrinth on coins as early as 430 BC, and similar non-branching patterns became widely used as visual representations of the Labyrinth – even though both logic and literary descriptions make it clear that the Minotaur was trapped in a complex branching maze.
Puts on sunglasses
A-maze-ing.
I think you need to be very careful in how you define a maze, lest you run the risk of rediscovering the difficulties involved with the Jordan Curve Theorem.