AdventOfCode/2024/12/12.md

## \-\-- Day 12: Garden Groups \-\--

Why not search for the Chief Historian near the [gardener](/2023/day/5)
and his [massive farm](/2023/day/21)? There\'s plenty of food, so The
Historians grab something to eat while they search.

You\'re about to settle near a complex arrangement of garden plots when
some Elves ask if you can lend a hand. They\'d like to set up
fences
around each region of garden plots, but they can\'t figure out how much
fence they need to order or how much it will cost. They hand you a map
(your puzzle input) of the garden plots.

Each garden plot grows only a single type of plant and is indicated by a
single letter on your map. When multiple garden plots are growing the
same type of plant and are touching (horizontally or vertically), they
form a *region*. For example:

    AAAA
    BBCD
    BBCC
    EEEC

This 4x4 arrangement includes garden plots growing five different types
of plants (labeled `A`, `B`, `C`, `D`, and `E`), each grouped into their
own region.

In order to accurately calculate the cost of the fence around a single
region, you need to know that region\'s *area* and *perimeter*.

The *area* of a region is simply the number of garden plots the region
contains. The above map\'s type `A`, `B`, and `C` plants are each in a
region of area `4`. The type `E` plants are in a region of area `3`; the
type `D` plants are in a region of area `1`.

Each garden plot is a square and so has *four sides*. The *perimeter* of
a region is the number of sides of garden plots in the region that do
not touch another garden plot in the same region. The type `A` and `C`
plants are each in a region with perimeter `10`. The type `B` and `E`
plants are each in a region with perimeter `8`. The lone `D` plot forms
its own region with perimeter `4`.

Visually indicating the sides of plots in each region that contribute to
the perimeter using `-` and `|`, the above map\'s regions\' perimeters
are measured as follows:

    +-+-+-+-+
    |A A A A|
    +-+-+-+-+     +-+
                  |D|
    +-+-+   +-+   +-+
    |B B|   |C|
    +   +   + +-+
    |B B|   |C C|
    +-+-+   +-+ +
              |C|
    +-+-+-+   +-+
    |E E E|
    +-+-+-+

Plants of the same type can appear in multiple separate regions, and
regions can even appear within other regions. For example:

    OOOOO
    OXOXO
    OOOOO
    OXOXO
    OOOOO

The above map contains *five* regions, one containing all of the `O`
garden plots, and the other four each containing a single `X` plot.

The four `X` regions each have area `1` and perimeter `4`. The region
containing `21` type `O` plants is more complicated; in addition to its
outer edge contributing a perimeter of `20`, its boundary with each `X`
region contributes an additional `4` to its perimeter, for a total
perimeter of `36`.

Due to \"modern\" business practices, the *price* of fence required for
a region is found by *multiplying* that region\'s area by its perimeter.
The *total price* of fencing all regions on a map is found by adding
together the price of fence for every region on the map.

In the first example, region `A` has price `4 * 10 = 40`, region `B` has
price `4 * 8 = 32`, region `C` has price `4 * 10 = 40`, region `D` has
price `1 * 4 = 4`, and region `E` has price `3 * 8 = 24`. So, the total
price for the first example is `140`.

In the second example, the region with all of the `O` plants has price
`21 * 36 = 756`, and each of the four smaller `X` regions has price
`1 * 4 = 4`, for a total price of `772` (`756 + 4 + 4 + 4 + 4`).

Here\'s a larger example:

    RRRRIICCFF
    RRRRIICCCF
    VVRRRCCFFF
    VVRCCCJFFF
    VVVVCJJCFE
    VVIVCCJJEE
    VVIIICJJEE
    MIIIIIJJEE
    MIIISIJEEE
    MMMISSJEEE

It contains:

-   A region of `R` plants with price `12 * 18 = 216`.
-   A region of `I` plants with price `4 * 8 = 32`.
-   A region of `C` plants with price `14 * 28 = 392`.
-   A region of `F` plants with price `10 * 18 = 180`.
-   A region of `V` plants with price `13 * 20 = 260`.
-   A region of `J` plants with price `11 * 20 = 220`.
-   A region of `C` plants with price `1 * 4 = 4`.
-   A region of `E` plants with price `13 * 18 = 234`.
-   A region of `I` plants with price `14 * 22 = 308`.
-   A region of `M` plants with price `5 * 12 = 60`.
-   A region of `S` plants with price `3 * 8 = 24`.

So, it has a total price of `1930`.

*What is the total price of fencing all regions on your map?*

Your puzzle answer was `1437300`.

The first half of this puzzle is complete! It provides one gold star: \*

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

Fortunately, the Elves are trying to order so much fence that they
qualify for a *bulk discount*!

Under the bulk discount, instead of using the perimeter to calculate the
price, you need to use the *number of sides* each region has. Each
straight section of fence counts as a side, regardless of how long it
is.

Consider this example again:

    AAAA
    BBCD
    BBCC
    EEEC

The region containing type `A` plants has `4` sides, as does each of the
regions containing plants of type `B`, `D`, and `E`. However, the more
complex region containing the plants of type `C` has `8` sides!

Using the new method of calculating the per-region price by multiplying
the region\'s area by its number of sides, regions `A` through `E` have
prices `16`, `16`, `32`, `4`, and `12`, respectively, for a total price
of `80`.

The second example above (full of type `X` and `O` plants) would have a
total price of `436`.

Here\'s a map that includes an E-shaped region full of type `E` plants:

    EEEEE
    EXXXX
    EEEEE
    EXXXX
    EEEEE

The E-shaped region has an area of `17` and `12` sides for a price of
`204`. Including the two regions full of type `X` plants, this map has a
total price of `236`.

This map has a total price of `368`:

    AAAAAA
    AAABBA
    AAABBA
    ABBAAA
    ABBAAA
    AAAAAA

It includes two regions full of type `B` plants (each with `4` sides)
and a single region full of type `A` plants (with `4` sides on the
outside and `8` more sides on the inside, a total of `12` sides). Be
especially careful when counting the fence around regions like the one
full of type `A` plants; in particular, each section of fence has an
in-side and an out-side, so the fence does not connect across the middle
of the region (where the two `B` regions touch diagonally). (The Elves
would have used the Möbius Fencing Company instead, but their contract
terms were too one-sided.)

The larger example from before now has the following updated prices:

-   A region of `R` plants with price `12 * 10 = 120`.
-   A region of `I` plants with price `4 * 4 = 16`.
-   A region of `C` plants with price `14 * 22 = 308`.
-   A region of `F` plants with price `10 * 12 = 120`.
-   A region of `V` plants with price `13 * 10 = 130`.
-   A region of `J` plants with price `11 * 12 = 132`.
-   A region of `C` plants with price `1 * 4 = 4`.
-   A region of `E` plants with price `13 * 8 = 104`.
-   A region of `I` plants with price `14 * 16 = 224`.
-   A region of `M` plants with price `5 * 6 = 30`.
-   A region of `S` plants with price `3 * 6 = 18`.

Adding these together produces its new total price of `1206`.

*What is the new total price of fencing all regions on your map?*

Answer:

Although it hasn\'t changed, you can still [get your puzzle
input](12/input).