## \-\-- Day 10: Hoof It \-\--

You all arrive at a [Lava Production Facility](/2023/day/15) on a
floating island in the sky. As the others begin to search the massive
industrial complex, you feel a small nose boop your leg and look down to
discover a reindeer wearing a hard
hat.

The reindeer is holding a book titled \"Lava Island Hiking Guide\".
However, when you open the book, you discover that most of it seems to
have been scorched by lava! As you\'re about to ask how you can help,
the reindeer brings you a blank [topographic
map](https://en.wikipedia.org/wiki/Topographic_map) of
the surrounding area (your puzzle input) and looks up at you excitedly.

Perhaps you can help fill in the missing hiking trails?

The topographic map indicates the *height* at each position using a
scale from `0` (lowest) to `9` (highest). For example:

    0123
    1234
    8765
    9876

Based on un-scorched scraps of the book, you determine that a good
hiking trail is *as long as possible* and has an *even, gradual, uphill
slope*. For all practical purposes, this means that a *hiking trail* is
any path that starts at height `0`, ends at height `9`, and always
increases by a height of exactly 1 at each step. Hiking trails never
include diagonal steps - only up, down, left, or right (from the
perspective of the map).

You look up from the map and notice that the reindeer has helpfully
begun to construct a small pile of pencils, markers, rulers, compasses,
stickers, and other equipment you might need to update the map with
hiking trails.

A *trailhead* is any position that starts one or more hiking trails -
here, these positions will always have height `0`. Assembling more
fragments of pages, you establish that a trailhead\'s *score* is the
number of `9`-height positions reachable from that trailhead via a
hiking trail. In the above example, the single trailhead in the top left
corner has a score of `1` because it can reach a single `9` (the one in
the bottom left).

This trailhead has a score of `2`:

    ...0...
    ...1...
    ...2...
    6543456
    7.....7
    8.....8
    9.....9

(The positions marked `.` are impassable tiles to simplify these
examples; they do not appear on your actual topographic map.)

This trailhead has a score of `4` because every `9` is reachable via a
hiking trail except the one immediately to the left of the trailhead:

    ..90..9
    ...1.98
    ...2..7
    6543456
    765.987
    876....
    987....

This topographic map contains *two* trailheads; the trailhead at the top
has a score of `1`, while the trailhead at the bottom has a score of
`2`:

    10..9..
    2...8..
    3...7..
    4567654
    ...8..3
    ...9..2
    .....01

Here\'s a larger example:

    89010123
    78121874
    87430965
    96549874
    45678903
    32019012
    01329801
    10456732

This larger example has 9 trailheads. Considering the trailheads in
reading order, they have scores of `5`, `6`, `5`, `3`, `1`, `3`, `5`,
`3`, and `5`. Adding these scores together, the sum of the scores of all
trailheads is `36`.

The reindeer gleefully carries over a protractor and adds it to the
pile. *What is the sum of the scores of all trailheads on your
topographic map?*

Your puzzle answer was `754`.

## \-\-- Part Two \-\-- {#part2}

The reindeer spends a few minutes reviewing your hiking trail map before
realizing something, disappearing for a few minutes, and finally
returning with yet another slightly-charred piece of paper.

The paper describes a second way to measure a trailhead called its
*rating*. A trailhead\'s rating is the *number of distinct hiking
trails* which begin at that trailhead. For example:

    .....0.
    ..4321.
    ..5..2.
    ..6543.
    ..7..4.
    ..8765.
    ..9....

The above map has a single trailhead; its rating is `3` because there
are exactly three distinct hiking trails which begin at that position:

    .....0.   .....0.   .....0.
    ..4321.   .....1.   .....1.
    ..5....   .....2.   .....2.
    ..6....   ..6543.   .....3.
    ..7....   ..7....   .....4.
    ..8....   ..8....   ..8765.
    ..9....   ..9....   ..9....

Here is a map containing a single trailhead with rating `13`:

    ..90..9
    ...1.98
    ...2..7
    6543456
    765.987
    876....
    987....

This map contains a single trailhead with rating `227` (because there
are `121` distinct hiking trails that lead to the `9` on the right edge
and `106` that lead to the `9` on the bottom edge):

    012345
    123456
    234567
    345678
    4.6789
    56789.

Here\'s the larger example from before:

    89010123
    78121874
    87430965
    96549874
    45678903
    32019012
    01329801
    10456732

Considering its trailheads in reading order, they have ratings of `20`,
`24`, `10`, `4`, `1`, `4`, `5`, `8`, and `5`. The sum of all trailhead
ratings in this larger example topographic map is `81`.

You\'re not sure how, but the reindeer seems to have crafted some tiny
flags out of toothpicks and bits of paper and is using them to mark
trailheads on your topographic map. *What is the sum of the ratings of
all trailheads?*

Your puzzle answer was `1609`.

Both parts of this puzzle are complete! They provide two gold stars:
\*\*

At this point, you should [return to your Advent calendar](/2024) and
try another puzzle.

If you still want to see it, you can [get your puzzle
input](10/input).