683 lines
13 KiB
Plaintext
683 lines
13 KiB
Plaintext
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--- Day 9: Rope Bridge ---
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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.
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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.
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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.
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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.
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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.
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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):
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....
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.TH.
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....
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....
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.H..
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..T.
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....
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...
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.H. (H covers T)
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...
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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:
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..... ..... .....
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.TH.. -> .T.H. -> ..TH.
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..... ..... .....
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... ... ...
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.T. .T. ...
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.H. -> ... -> .T.
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... .H. .H.
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... ... ...
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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:
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..... ..... .....
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..... ..H.. ..H..
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..H.. -> ..... -> ..T..
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.T... .T... .....
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..... ..... .....
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..... ..... .....
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..... ..... .....
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..H.. -> ...H. -> ..TH.
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.T... .T... .....
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..... ..... .....
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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.
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For example:
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R 4
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U 4
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L 3
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D 1
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R 4
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D 1
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L 5
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R 2
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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):
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== Initial State ==
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......
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......
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......
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......
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H..... (H covers T, s)
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== R 4 ==
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......
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......
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......
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......
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TH.... (T covers s)
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......
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......
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......
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......
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sTH...
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......
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......
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......
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......
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s.TH..
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......
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......
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......
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......
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s..TH.
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== U 4 ==
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......
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......
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......
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....H.
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s..T..
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......
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......
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....H.
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....T.
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s.....
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......
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....H.
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....T.
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......
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s.....
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....H.
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....T.
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......
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......
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s.....
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== L 3 ==
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...H..
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....T.
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......
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......
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s.....
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..HT..
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......
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......
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......
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s.....
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.HT...
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......
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......
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......
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s.....
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== D 1 ==
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..T...
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.H....
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......
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......
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s.....
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== R 4 ==
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..T...
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..H...
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......
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......
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s.....
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..T...
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...H..
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......
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......
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s.....
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......
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...TH.
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......
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......
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s.....
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......
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....TH
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......
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......
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s.....
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== D 1 ==
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......
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....T.
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.....H
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......
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s.....
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== L 5 ==
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......
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....T.
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....H.
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......
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s.....
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......
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....T.
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...H..
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......
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s.....
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......
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......
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..HT..
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......
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s.....
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......
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......
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.HT...
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......
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s.....
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......
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......
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HT....
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......
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s.....
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== R 2 ==
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......
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......
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.H.... (H covers T)
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......
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s.....
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......
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......
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.TH...
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......
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s.....
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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:
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..##..
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...##.
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.####.
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....#.
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s###..
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So, there are 13 positions the tail visited at least once.
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Simulate your complete hypothetical series of motions. How many positions does the tail of the rope visit at least once?
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--- Part Two ---
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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!
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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.
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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.
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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:
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== Initial State ==
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......
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......
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......
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......
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H..... (H covers 1, 2, 3, 4, 5, 6, 7, 8, 9, s)
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== R 4 ==
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......
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......
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......
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......
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1H.... (1 covers 2, 3, 4, 5, 6, 7, 8, 9, s)
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......
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......
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......
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......
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21H... (2 covers 3, 4, 5, 6, 7, 8, 9, s)
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......
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......
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......
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......
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321H.. (3 covers 4, 5, 6, 7, 8, 9, s)
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......
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......
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......
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......
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4321H. (4 covers 5, 6, 7, 8, 9, s)
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== U 4 ==
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......
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......
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......
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....H.
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4321.. (4 covers 5, 6, 7, 8, 9, s)
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......
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......
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....H.
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.4321.
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5..... (5 covers 6, 7, 8, 9, s)
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......
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....H.
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....1.
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.432..
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5..... (5 covers 6, 7, 8, 9, s)
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....H.
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....1.
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..432.
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.5....
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6..... (6 covers 7, 8, 9, s)
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== L 3 ==
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...H..
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....1.
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..432.
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.5....
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6..... (6 covers 7, 8, 9, s)
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..H1..
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...2..
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..43..
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.5....
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6..... (6 covers 7, 8, 9, s)
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.H1...
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...2..
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..43..
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.5....
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6..... (6 covers 7, 8, 9, s)
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== D 1 ==
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..1...
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.H.2..
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..43..
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.5....
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6..... (6 covers 7, 8, 9, s)
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== R 4 ==
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..1...
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..H2..
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..43..
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.5....
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6..... (6 covers 7, 8, 9, s)
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..1...
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...H.. (H covers 2)
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..43..
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.5....
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6..... (6 covers 7, 8, 9, s)
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......
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...1H. (1 covers 2)
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..43..
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.5....
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6..... (6 covers 7, 8, 9, s)
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......
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...21H
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..43..
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.5....
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6..... (6 covers 7, 8, 9, s)
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== D 1 ==
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......
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...21.
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..43.H
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.5....
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6..... (6 covers 7, 8, 9, s)
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== L 5 ==
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......
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...21.
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..43H.
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.5....
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6..... (6 covers 7, 8, 9, s)
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......
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...21.
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..4H.. (H covers 3)
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.5....
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6..... (6 covers 7, 8, 9, s)
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......
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...2..
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..H1.. (H covers 4; 1 covers 3)
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.5....
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6..... (6 covers 7, 8, 9, s)
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......
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...2..
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.H13.. (1 covers 4)
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.5....
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6..... (6 covers 7, 8, 9, s)
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......
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......
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H123.. (2 covers 4)
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.5....
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6..... (6 covers 7, 8, 9, s)
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== R 2 ==
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......
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......
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.H23.. (H covers 1; 2 covers 4)
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.5....
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6..... (6 covers 7, 8, 9, s)
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......
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......
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.1H3.. (H covers 2, 4)
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.5....
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6..... (6 covers 7, 8, 9, s)
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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.
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Here's a larger example:
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R 5
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U 8
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L 8
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D 3
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R 17
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D 10
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L 25
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U 20
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These motions occur as follows (individual steps are not shown):
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== Initial State ==
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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...........H.............. (H covers 1, 2, 3, 4, 5, 6, 7, 8, 9, s)
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..........................
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..........................
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..........................
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..........................
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== R 5 ==
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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...........54321H......... (5 covers 6, 7, 8, 9, s)
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..........................
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..........................
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..........................
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..........................
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== U 8 ==
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..........................
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..........................
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..........................
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................H.........
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................1.........
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................2.........
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................3.........
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...............54.........
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..............6...........
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.............7............
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............8.............
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...........9.............. (9 covers s)
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..........................
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..........................
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..........................
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== L 8 ==
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..........................
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..........................
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........H1234.............
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............5.............
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............6.............
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............7.............
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............8.............
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............9.............
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..........................
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..........................
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...........s..............
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..........................
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..........................
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..........................
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..........................
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== D 3 ==
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
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.........2345.............
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........1...6.............
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........H...7.............
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............8.............
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............9.............
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..........................
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..........................
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...........s..............
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..........................
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..........................
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..........................
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..........................
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..........................
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== R 17 ==
|
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..........................
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..........................
|
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..........................
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..........................
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..........................
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..........................
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..........................
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..........................
|
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..........................
|
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..........................
|
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................987654321H
|
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..........................
|
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..........................
|
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..........................
|
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..........................
|
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...........s..............
|
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..........................
|
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..........................
|
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..........................
|
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..........................
|
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..........................
|
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|
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|
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== D 10 ==
|
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|
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..........................
|
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..........................
|
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..........................
|
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..........................
|
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..........................
|
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..........................
|
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|
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..........................
|
||
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..........................
|
||
|
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..........................
|
||
|
|
..........................
|
||
|
|
..........................
|
||
|
|
..........................
|
||
|
|
..........................
|
||
|
|
..........................
|
||
|
|
..........................
|
||
|
|
...........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?
|