Add day 18 explanations (#496)

* Add day18 explanations

* Update docs/2023/puzzles/day18.md

---------

Co-authored-by: Jamie Thompson <[email protected]>

Author: EugeneFlesselle

docs/2023/puzzles/day18.md

@@ -2,10 +2,136 @@ import Solver from "../../../../../website/src/components/Solver.js"
 
 # Day 18: Lavaduct Lagoon
 
+by [@EugeneFlesselle](https://github.com/EugeneFlesselle)
+
 ## Puzzle description
 
 https://adventofcode.com/2023/day/18
 
+## Solution Summary
+
+Assume we have a given `digPlan: Seq[Trench]` for which to compute the area,
+and let the following classes:
+```scala 3
+enum Direction:
+  case Up, Down, Left, Right
+
+case class Trench(dir: Direction, length: Int)
+```
+
+We can go through the dig plan keeping track of the current position,
+by starting from `(x = 0, y = 0)`, increasing `x` when going right, increasing `y` when going down, and so on.
+
+Provided our current position, we can then keep track of the lagoon area as follows:
+- When going `Right`: we count all the points in the line we cover, i.e the `length` of the trench.
+- When going `Down`: we count all the points which we leave on the left (or pass over),
+  i.e. the `length` of the downwards trench times our current `x` coordinate,
+  `+1` to count the current vertical trench as in the area.
+  Of course, we may be including too much at this point,
+  since we do not yet know what part of the left is actually in the lagoon,
+  but we will account for it later.
+- When going `Left`: there is nothing to add,
+  the position could only have been reached from a downwards trench,
+  hence the area has already been counted.
+- When going `Up`: we now know by how much we had over increased the area when going down,
+  and can remove everything strictly to the left of the current `x` position.
+
+In summary, we assume we cover everything to the left when going down
+and remove the uncovered part when coming back up.
+Finally, we must start from an area of `1`as the starting position `(0, 0)` is naturally covered,
+but isn't counted by the first trench whichever it may be.
+All of which translates to the following foldLeft in scala 😉:
+
+```scala 3
+val (_, area) = digPlan.foldLeft((0, 0), 1L):
+  case (((x, y), area), Trench(dir, len)) => dir match
+    case Right => ((x + len, y), area + len)
+    case Down  => ((x, y + len), area + (x + 1) * len.toLong)
+    case Left  => ((x - len, y), area)
+    case Up    => ((x, y - len), area - x * len.toLong)
+```
+
+Also note we have to use `Long`s to avoid the computations from overflowing.
+
+
+### Part 1
+
+We can get the `Trench` of each line in the `input: String`,
+by parsing the direction from the corresponding character
+and ignoring the color of the trench.
+And then proceed as above with the obtained `digPlan`.
+
+```scala 3
+object Direction:
+  def fromChar(c: Char): Direction = c match
+    case 'U' => Up case 'D' => Down case 'L' => Left case 'R' => Right
+
+val digPlan = for
+  case s"$dirC $len (#$_)" <- input.linesIterator
+  dir = Direction.fromChar(dirC.head)
+yield Trench(dir, len.toInt)
+```
+
+### Part 2
+
+We do the same for part 2, except we use the color to
+get the trench fields: 
+its direction from the last digit,
+and its length by converting from the hexadecimal encoding of the remaining digits.
+
+```scala 3
+object Direction:
+  def fromInt(i: Char): Direction = i match
+    case '0' => Right case '1' => Down case '2' => Left case '3' => Up
+
+val digPlan = for
+  case s"$_ $_ (#$color)" <- input.linesIterator
+  dir = Direction.fromInt(color.last)
+  len = BigInt(x = color.init, radix = 16)
+yield Trench(dir, len.toInt)
+```
+
+## Final code
+
+```scala 3
+enum Direction:
+  case Up, Down, Left, Right
+object Direction:
+  def fromChar(c: Char): Direction = c match
+    case 'U' => Up case 'D' => Down case 'L' => Left case 'R' => Right
+  def fromInt(i: Char): Direction = i match
+    case '0' => Right case '1' => Down case '2' => Left case '3' => Up
+import Direction.*
+
+case class Trench(dir: Direction, length: Int)
+
+def area(digPlan: Seq[Trench]): Long =
+  val (_, area) = digPlan.foldLeft((0, 0), 1L):
+    case (((x, y), area), Trench(dir, len)) => dir match
+      case Right => ((x + len, y), area + len)
+      case Down  => ((x, y + len), area + (x + 1) * len.toLong)
+      case Left  => ((x - len, y), area)
+      case Up    => ((x, y - len), area - x * len.toLong)
+  area
+
+def part1(input: String): String =
+  val digPlan = for
+    case s"$dirC $len (#$_)" <- input.linesIterator
+    dir = Direction.fromChar(dirC.head)
+  yield Trench(dir, len.toInt)
+  
+  area(digPlan.toSeq).toString
+
+def part2(input: String): String =
+  val digPlan = for
+    case s"$_ $_ (#$color)" <- input.linesIterator
+    dir = Direction.fromInt(color.last)
+    len = BigInt(x = color.init, radix = 16)
+  yield Trench(dir, len.toInt)
+
+  area(digPlan.toSeq).toString
+```
+
 ## Solutions from the community
 
 - [Solution](https://github.com/merlinorg/aoc2023/blob/main/src/main/scala/Day18.scala) by [merlin](https://github.com/merlinorg/)