day9

Day 9: Rope Bridge

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Part 1

This rope bridge creaks as you walk along it. You aren't sure how old it is, or whether it can even support your weight.

It seems to support the Elves just fine, though. The bridge spans a gorge which was carved out by the massive river far below you.

You step carefully; as you do, the ropes stretch and twist. You decide to distract yourself by modeling rope physics; maybe you can even figure out where not to step.

Consider a rope with a knot at each end; these knots mark the head and the tail of the rope. If the head moves far enough away from the tail, the tail is pulled toward the head.

Due to nebulous reasoning involving Planck lengths, you should be able to model the positions of the knots on a two-dimensional grid. Then, by following a hypothetical series of motions (your puzzle input) for the head, you can determine how the tail will move.

Due to the aforementioned Planck lengths, the rope must be quite short; in fact, the head (H) and tail (T) must always be touching (diagonally adjacent and even overlapping both count as touching):

....
.TH.
....

....
.H..
..T.
....

...
.H. (H covers T)
...

If the head is ever two steps directly up, down, left, or right from the tail, the tail must also move one step in that direction so it remains close enough:

.....    .....    .....
.TH.. -> .T.H. -> ..TH.
.....    .....    .....

...    ...    ...
.T.    .T.    ...
.H. -> ... -> .T.
...    .H.    .H.
...    ...    ...

Otherwise, if the head and tail aren't touching and aren't in the same row or column, the tail always moves one step diagonally to keep up:

.....    .....    .....
.....    ..H..    ..H..
..H.. -> ..... -> ..T..
.T...    .T...    .....
.....    .....    .....

.....    .....    .....
.....    .....    .....
..H.. -> ...H. -> ..TH.
.T...    .T...    .....
.....    .....    .....

You just need to work out where the tail goes as the head follows a series of motions. Assume the head and the tail both start at the same position, overlapping.

For example:

R 4
U 4
L 3
D 1
R 4
D 1
L 5
R 2

This series of motions moves the head right four steps, then up four steps, then left three steps, then down one step, and so on. After each step, you'll need to update the position of the tail if the step means the head is no longer adjacent to the tail. Visually, these motions occur as follows (s marks the starting position as a reference point):

== Initial State ==

......
......
......
......
H.....  (H covers T, s)

== R 4 ==

......
......
......
......
TH....  (T covers s)

......
......
......
......
sTH...

......
......
......
......
s.TH..

......
......
......
......
s..TH.

== U 4 ==

......
......
......
....H.
s..T..

......
......
....H.
....T.
s.....

......
....H.
....T.
......
s.....

....H.
....T.
......
......
s.....

== L 3 ==

...H..
....T.
......
......
s.....

..HT..
......
......
......
s.....

.HT...
......
......
......
s.....

== D 1 ==

..T...
.H....
......
......
s.....

== R 4 ==

..T...
..H...
......
......
s.....

..T...
...H..
......
......
s.....

......
...TH.
......
......
s.....

......
....TH
......
......
s.....

== D 1 ==

......
....T.
.....H
......
s.....

== L 5 ==

......
....T.
....H.
......
s.....

......
....T.
...H..
......
s.....

......
......
..HT..
......
s.....

......
......
.HT...
......
s.....

......
......
HT....
......
s.....

== R 2 ==

......
......
.H....  (H covers T)
......
s.....

......
......
.TH...
......
s.....

After simulating the rope, you can count up all of the positions the tail visited at least once. In this diagram, s again marks the starting position (which the tail also visited) and # marks other positions the tail visited:

..##..
...##.
.####.
....#.
s###..

So, there are 13 positions the tail visited at least once.

Simulate your complete hypothetical series of motions. How many positions does the tail of the rope visit at least once?

Solution

Kind of convoluted. I created a struct to keep track of things. The struct has

• [2]int for the current tail and head coordinates
• map[int]struct{} to keep track of where the tail has been, and
• a logger

For helper functions there's one called coordToBinary which turns an x,y coordinate into one number by adding 512 to both (so as to not deal with negative numbers), and then shifting the x coordinate by 10 bits, and then oring them.

There's another one for visualisations: those are useful for testing / debugging.

And a third for parsing the commands: what does "U 5" mean?

Basic logic:

• for each line of the commands
  • figure out the direction, and how much distance to move
  • pass that onto the moveHead method on the rope
  • depending on the direction a unit coordinate pair is selected
  • and then a for loop happens for the distance
    • within each loop the head is moved by the unit coordinate pair
    • then a check happens to see whether the head-tail distance is big enough for the tail to move with it
      • if it is, all 16 possible locations of the tail with relation to the head are considered, and depending on that, a unit coordinate is chosen
      • the tail is moved by that coordinate pair
      • its location is recorded in the map
• at the end of it we read the lenght of the tail trace map, and that is our solution

Part 2

A rope snaps! Suddenly, the river is getting a lot closer than you remember. The bridge is still there, but some of the ropes that broke are now whipping toward you as you fall through the air!

The ropes are moving too quickly to grab; you only have a few seconds to choose how to arch your body to avoid being hit. Fortunately, your simulation can be extended to support longer ropes.

Rather than two knots, you now must simulate a rope consisting of ten knots. One knot is still the head of the rope and moves according to the series of motions. Each knot further down the rope follows the knot in front of it using the same rules as before.

Using the same series of motions as the above example, but with the knots marked H, 1, 2, ..., 9, the motions now occur as follows:

== Initial State ==

......
......
......
......
H.....  (H covers 1, 2, 3, 4, 5, 6, 7, 8, 9, s)

== R 4 ==

......
......
......
......
1H....  (1 covers 2, 3, 4, 5, 6, 7, 8, 9, s)

......
......
......
......
21H...  (2 covers 3, 4, 5, 6, 7, 8, 9, s)

......
......
......
......
321H..  (3 covers 4, 5, 6, 7, 8, 9, s)

......
......
......
......
4321H.  (4 covers 5, 6, 7, 8, 9, s)

== U 4 ==

......
......
......
....H.
4321..  (4 covers 5, 6, 7, 8, 9, s)

......
......
....H.
.4321.
5.....  (5 covers 6, 7, 8, 9, s)

......
....H.
....1.
.432..
5.....  (5 covers 6, 7, 8, 9, s)

....H.
....1.
..432.
.5....
6.....  (6 covers 7, 8, 9, s)

== L 3 ==

...H..
....1.
..432.
.5....
6.....  (6 covers 7, 8, 9, s)

..H1..
...2..
..43..
.5....
6.....  (6 covers 7, 8, 9, s)

.H1...
...2..
..43..
.5....
6.....  (6 covers 7, 8, 9, s)

== D 1 ==

..1...
.H.2..
..43..
.5....
6.....  (6 covers 7, 8, 9, s)

== R 4 ==

..1...
..H2..
..43..
.5....
6.....  (6 covers 7, 8, 9, s)

..1...
...H..  (H covers 2)
..43..
.5....
6.....  (6 covers 7, 8, 9, s)

......
...1H.  (1 covers 2)
..43..
.5....
6.....  (6 covers 7, 8, 9, s)

......
...21H
..43..
.5....
6.....  (6 covers 7, 8, 9, s)

== D 1 ==

......
...21.
..43.H
.5....
6.....  (6 covers 7, 8, 9, s)

== L 5 ==

......
...21.
..43H.
.5....
6.....  (6 covers 7, 8, 9, s)

......
...21.
..4H..  (H covers 3)
.5....
6.....  (6 covers 7, 8, 9, s)

......
...2..
..H1..  (H covers 4; 1 covers 3)
.5....
6.....  (6 covers 7, 8, 9, s)

......
...2..
.H13..  (1 covers 4)
.5....
6.....  (6 covers 7, 8, 9, s)

......
......
H123..  (2 covers 4)
.5....
6.....  (6 covers 7, 8, 9, s)

== R 2 ==

......
......
.H23..  (H covers 1; 2 covers 4)
.5....
6.....  (6 covers 7, 8, 9, s)

......
......
.1H3..  (H covers 2, 4)
.5....
6.....  (6 covers 7, 8, 9, s)

Now, you need to keep track of the positions the new tail, 9, visits. In this example, the tail never moves, and so it only visits 1 position. However, be careful: more types of motion are possible than before, so you might want to visually compare your simulated rope to the one above.

Here's a larger example:

R 5
U 8
L 8
D 3
R 17
D 10
L 25
U 20

These motions occur as follows (individual steps are not shown):

== Initial State ==

..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
...........H..............  (H covers 1, 2, 3, 4, 5, 6, 7, 8, 9, s)
..........................
..........................
..........................
..........................
..........................

== R 5 ==

..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
...........54321H.........  (5 covers 6, 7, 8, 9, s)
..........................
..........................
..........................
..........................
..........................

== U 8 ==

..........................
..........................
..........................
..........................
..........................
..........................
..........................
................H.........
................1.........
................2.........
................3.........
...............54.........
..............6...........
.............7............
............8.............
...........9..............  (9 covers s)
..........................
..........................
..........................
..........................
..........................

== L 8 ==

..........................
..........................
..........................
..........................
..........................
..........................
..........................
........H1234.............
............5.............
............6.............
............7.............
............8.............
............9.............
..........................
..........................
...........s..............
..........................
..........................
..........................
..........................
..........................

== D 3 ==

..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
.........2345.............
........1...6.............
........H...7.............
............8.............
............9.............
..........................
..........................
...........s..............
..........................
..........................
..........................
..........................
..........................

== R 17 ==

..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
................987654321H
..........................
..........................
..........................
..........................
...........s..............
..........................
..........................
..........................
..........................
..........................

== D 10 ==

..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
...........s.........98765
.........................4
.........................3
.........................2
.........................1
.........................H

== L 25 ==

..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
...........s..............
..........................
..........................
..........................
..........................
H123456789................

== U 20 ==

H.........................
1.........................
2.........................
3.........................
4.........................
5.........................
6.........................
7.........................
8.........................
9.........................
..........................
..........................
..........................
..........................
..........................
...........s..............
..........................
..........................
..........................
..........................
..........................

Now, the tail (9) visits 36 positions (including s) at least once:

..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
..........................
#.........................
#.............###.........
#............#...#........
.#..........#.....#.......
..#..........#.....#......
...#........#.......#.....
....#......s.........#....
.....#..............#.....
......#............#......
.......#..........#.......
........#........#........
.........########.........

Simulate your complete series of motions on a larger rope with ten knots. How many positions does the tail of the rope visit at least once?

Solution

Mostly the same as above, except having two separate properties for the coordinates fo the head and tail, now there's one 10-long array of coordinates.

• the head is moved as above, and then it calls a moveChain method
• that method will loop through all the links from the head (index 0) up until the penultimate link (the one before the tail), and pass that, and the next link, onto a moveNext method
  • for each link pair it checks if the next link needs to be moved. The logic is the same as "should we move the tail" in part 1
  • if the next link had to be moved, the unit coordinate is returned, if not, a 0,0 coordinate pair returned
  • if at any point moveNext returns a 0,0 pair, we break the chain, because if chain 4 didn't move, then 5, 6, 7, 8, 9 will definitely not need to move either
  • if the last link moves (index 9), we record the new location to the map
• at the end we read the length of the trace map