https://adventofcode.com/2021/day/12

## \--- Day 12: Passage Pathing ---

With your submarine's subterranean subsystems subsisting suboptimally, the
only way you're getting out of this cave anytime soon is by finding a path
yourself. Not just _a_ path - the only way to know if you've found the _best_
path is to find _all_ of them.

Fortunately, the sensors are still mostly working, and so you build a rough
map of the remaining caves (your puzzle input). For example:

[code]

    start-A
    start-b
    A-c
    A-b
    b-d
    A-end
    b-end
    
[/code]

This is a list of how all of the caves are connected. You start in the cave
named `start`, and your destination is the cave named `end`. An entry like
`b-d` means that cave `b` is connected to cave `d` \- that is, you can move
between them.

So, the above cave system looks roughly like this:

[code]

        start
        /   \
    c--A-----b--d
        \   /
         end
    
[/code]

Your goal is to find the number of distinct _paths_ that start at `start`, end
at `end`, and don't visit small caves more than once. There are two types of
caves: _big_ caves (written in uppercase, like `A`) and _small_ caves (written
in lowercase, like `b`). It would be a waste of time to visit any small cave
more than once, but big caves are large enough that it might be worth visiting
them multiple times. So, all paths you find should _visit small caves at most
once_ , and can _visit big caves any number of times_.

Given these rules, there are `_10_` paths through this example cave system:

[code]

    start,A,b,A,c,A,end
    start,A,b,A,end
    start,A,b,end
    start,A,c,A,b,A,end
    start,A,c,A,b,end
    start,A,c,A,end
    start,A,end
    start,b,A,c,A,end
    start,b,A,end
    start,b,end
    
[/code]

(Each line in the above list corresponds to a single path; the caves visited
by that path are listed in the order they are visited and separated by
commas.)

Note that in this cave system, cave `d` is never visited by any path: to do
so, cave `b` would need to be visited twice (once on the way to cave `d` and a
second time when returning from cave `d`), and since cave `b` is small, this
is not allowed.

Here is a slightly larger example:

[code]

    dc-end
    HN-start
    start-kj
    dc-start
    dc-HN
    LN-dc
    HN-end
    kj-sa
    kj-HN
    kj-dc
    
[/code]

The `19` paths through it are as follows:

[code]

    start,HN,dc,HN,end
    start,HN,dc,HN,kj,HN,end
    start,HN,dc,end
    start,HN,dc,kj,HN,end
    start,HN,end
    start,HN,kj,HN,dc,HN,end
    start,HN,kj,HN,dc,end
    start,HN,kj,HN,end
    start,HN,kj,dc,HN,end
    start,HN,kj,dc,end
    start,dc,HN,end
    start,dc,HN,kj,HN,end
    start,dc,end
    start,dc,kj,HN,end
    start,kj,HN,dc,HN,end
    start,kj,HN,dc,end
    start,kj,HN,end
    start,kj,dc,HN,end
    start,kj,dc,end
    
[/code]

Finally, this even larger example has `226` paths through it:

[code]

    fs-end
    he-DX
    fs-he
    start-DX
    pj-DX
    end-zg
    zg-sl
    zg-pj
    pj-he
    RW-he
    fs-DX
    pj-RW
    zg-RW
    start-pj
    he-WI
    zg-he
    pj-fs
    start-RW
    
[/code]

_How many paths through this cave system are there that visit small caves at
most once?_