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My puzzle as far as i know, corrections welcome..
Since the maze's structure is not specified, then
a random treatment is in order. In this way the path you
take at any point is a random variable.
There are two possibilities in the long term:
- you escape from the maze
- you are trapped forever in an infinite loop
Case 2 is a consequence of the strategy. More precisely,
if any point during your journey you take the same
path twice, you will be stuck in a loop forever. moreover,
if you take a certain path twice, it follows that youre
previous path was also taken twice, and so on until the
first path you took from the beginning.
The 2 possibilities are then:
- you escape
- you come back to your initial path
Which can be rewritten as
- you find the exit path
- you find your first path
Where the exit path is like any other path except it connects
a junction to the exit. Now, consider that you have already
taken 1 path (this is your first path). Each possibility above
is clearly equally likely, since each is represented by 1 path.
So, after your first path, there is 1/2 chance you'll escape and
1/2 chance youll be stuck forever. At the start,
there is 0 chance that youll retake your first path and
1/n chance that youll take the exit path, where n
is the total number of paths. So
P(escape) = (1/n * 1) + n-1/n * 1/2
P(trapped) = (1/n * 0) + n-1/n * 1/2
P(trapped) = 1 - P(escape)
n is approximately 3 * the number of junctions. as
the maze size increases the first term is neglibible
and P(trapped) = P(escape) = 1/2
So the best answer to the question is:
"There is equal chance that youll either escape or be trapped forever." |
