[FRIAM] solving mazes

[Roger Critchlow]

Okay Nick,

So you draw the maze as a graph where the nodes of the graph are positions
in the maze that can be occupied, and the edges of the graph are one step
open paths between adjacent positions.  Easy to draw on graph paper, here
is a site which generates graph paper pdfs for printing:
https://incompetech.com/graphpaper/, just fill in the cells that are walls
and draw lines to connect the remaining open cells.  Each open cell is a
node, each line connecting adjacent cells is an edge.

The graph of the maze must be connected or parts of the maze will be cut
off from each other and no algorithm will ever work if it starts in the
wrong connected component.  Nasty trick to play on a rat.

If the graph of the maze contains loops, then wall following will get
trapped in the loops, in fact, I think each loop defines a wall following
domain, and to solve the maze you'll have to detect when you've completed a
loop and switch to following the opposite wall when it's one you haven't
already followed.  If you just switch to the opposite wall at random,
you'll most probably end up on the same loop going in the opposite
direction.

Just to demonstrate how graph properties end up trapping the unwary, an
open courtyard in your maze ends up being an area of fully connected graph,
which is all loops all the time, which is a counter example to my proposal
that "each loop defines a wall following domain" since lots of the loops in
the courtyard will have no adjacent walls at all.  So I must have meant
something subtler than loop, something that is a loop that has walls on
both sides.

-- rec --

On Sun, Feb 28, 2021 at 9:39 AM Jochen Fromm <jofr at cas-group.net> wrote:

> Jamis Buck is very interested in mazes. He even wrote a book about it
> named "Mazes for programmers"
> https://www.jamisbuck.org/mazes/
>
> It contains algorithms for generating and solving mazes
> https://pragprog.com/titles/jbmaze/mazes-for-programmers/
>
> -J.
>
>
> -------- Original message --------
> From: cody dooderson <d00d3rs0n at gmail.com>
> Date: 2/28/21 01:10 (GMT+01:00)
> To: The Friday Morning Applied Complexity Coffee Group <friam at redfish.com>
>
> Subject: Re: [FRIAM] solving mazes
>
> I am assuming this is a 2D maze. Wikipedia does a better job at explaining
> the problems with wall following than I can.
>
> *If the maze is not simply-connected (i.e. if the start or endpoints are
> in the center of the structure surrounded by passage loops, or the pathways
> cross over and under each other and such parts of the solution path are
> surrounded by passage loops), this method will not reach the goal.*
>
>
>
>
> On Sat, Feb 27, 2021, 1:48 PM <thompnickson2 at gmail.com> wrote:
>
>> Hi, All,
>>
>> Due to a review I have been working on, I have been dragged back into
>> thinking about maze learning in rats.  Any animal I have ever known, when
>> confined, will explore the boundaries of its enclosure.  Cows, for instance
>> will beat a path just inside the barbed wire that encloses them.  So a maze
>> is not only a series of pathways but it is also an enclosure.  If the rat
>> puts his left whisker against the left wall of the maze, he will eventually
>> get to the goal box, right.  It works with the Hampton Court Maze.  On the
>> second run, he can now use odor cues, such that any time he encounters his
>> own odor both entering and leaving a passage way, he should just skip that
>> passage way.
>>
>> So I am wondering, you topologists (??) out there, how general is the
>> statement, “every maze is an enclosure”  and what is the limitation on the
>> idea that any maze can be solved by putting your right or left hand on a
>> wall and continuing to walk until you find the goal or are let out of the
>> maze.  Now I should quickly say that rat mazes are usually composed of a
>> series of bifurcating choice points, where the rat can go either left or
>> right. Some choices lead ultimately to dead ends.  In sum, a runway in such
>> a maze can go straight, turn R or L without choice or form a T with a right
>> or left choice.  My intuition is that no such maze can be designed that
>> does not permit the boundary following strategy.
>>
>> Nick
>>
>>
>>
>> Nick Thompson
>>
>> ThompNickSon2 at gmail.com
>>
>> https://wordpress.clarku.edu/nthompson/
>>
>>
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