- #1

DaveC426913

Gold Member

- 22,473

- 6,141

- TL;DR Summary
- Is there a mathematical subdiscipline of the study of mazes?

Some say you can escape a maze by placing your hand on one wall and tracing out the entire maze to escape.

But that only works if you manage to put your hand on an outside wall - a wall that is integral to the perimeter wall.

Some mazes have walls that are entirely interior - islands of walls. Put your hands on one of these and you will go around in a circle until/unless you realize you're retracing your steps.

My brother traversed a hedge maze the other day, and claimed he used the trick to find the exit, but I have my doubts it was a valid traversal.

He says he started at the single entrance/exit and traced his way all the way around to the same to escape.

I say, if a maze has a single entrance/exit, then the goal of solving such a maze is not to "find the exit", since that's literally trivial; the goal is (usually) to find the central courtyard (where the proverbial piece of cheese is waiting for the rat to find it).

Indeed, the maze solved has such a central courtyard:

And he would

The above maze has quite a few "islands", and I have realized that that is equivalent to saying the maze has multiple routes.

An island is ... mazologically ... the same as saying there are two possible paths.

I have realized that you could instantly see what kind of maze you have if you could sort of use it as a mold - pour a fluid into it (like thick jell-O) that, when set, you can pick up out of the maze by its entrance and exit. All dead ends would hang down, leaving you holding a valid path from one end to the other. Multiple paths would manifest as closed loops; the "islands" being the voids within the loops.

I also digitized the maze

to make an estimate on how likely one could find the exit if dropped into the maze (in the dark) at a random location.

I decided

By summing all diagonal moves that end up on an outside wall (206) - and dividing by the total possibilities (15x15*x4 = 900), I arrived at a solvability index for this maze of 22.9%. Thus,

*you count the

I am making all this up on my own (and possibly reinventing the wheel) because

But that only works if you manage to put your hand on an outside wall - a wall that is integral to the perimeter wall.

Some mazes have walls that are entirely interior - islands of walls. Put your hands on one of these and you will go around in a circle until/unless you realize you're retracing your steps.

My brother traversed a hedge maze the other day, and claimed he used the trick to find the exit, but I have my doubts it was a valid traversal.

He says he started at the single entrance/exit and traced his way all the way around to the same to escape.

I say, if a maze has a single entrance/exit, then the goal of solving such a maze is not to "find the exit", since that's literally trivial; the goal is (usually) to find the central courtyard (where the proverbial piece of cheese is waiting for the rat to find it).

Indeed, the maze solved has such a central courtyard:

And he would

*not*have found it by placing his hand on an outside wall.The above maze has quite a few "islands", and I have realized that that is equivalent to saying the maze has multiple routes.

An island is ... mazologically ... the same as saying there are two possible paths.

I have realized that you could instantly see what kind of maze you have if you could sort of use it as a mold - pour a fluid into it (like thick jell-O) that, when set, you can pick up out of the maze by its entrance and exit. All dead ends would hang down, leaving you holding a valid path from one end to the other. Multiple paths would manifest as closed loops; the "islands" being the voids within the loops.

I also digitized the maze

to make an estimate on how likely one could find the exit if dropped into the maze (in the dark) at a random location.

I decided

*every coordinate has no more than four meaningful directions one might choose*(the four diagonals of every coord). A "4" indicates you are guaranteed to hit an outside wall, no matter which direction you move to find a wall. A "0" indicates you are guaranteed to hit an island.By summing all diagonal moves that end up on an outside wall (206) - and dividing by the total possibilities (15x15*x4 = 900), I arrived at a solvability index for this maze of 22.9%. Thus,

*randomly dropped in, and then randomly picking a direction, you have a 22.9% chance of reaching the exit.**you count the

*paths*, not the*hedges:*a 16x16 hedge maze gets you 15x15 rows of pathsI am making all this up on my own (and possibly reinventing the wheel) because

**I have yet find the subdiscipline of math that defines this ... mazology. Suggestions?**
Last edited: