746 lines
13 KiB
Org Mode
746 lines
13 KiB
Org Mode
** --- Day 9: Rope Bridge ---
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This rope bridge creaks as you walk along it. You aren't sure how old it
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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
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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
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to distract yourself by modeling rope physics; maybe you can even figure
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out where /not/ to step.
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Consider a rope with a knot at each end; these knots mark the /head/ and
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the /tail/ of the rope. If the head moves far enough away from the tail,
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the tail is pulled toward the head.
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Due to nebulous reasoning involving
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[[https://en.wikipedia.org/wiki/Planck_units#Planck_length][Planck
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lengths]], you should be able to model the positions of the knots on a
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two-dimensional grid. Then, by following a hypothetical /series of
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motions/ (your puzzle input) for the head, you can determine how the
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tail will move.
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Due to the aforementioned Planck lengths, the rope must be quite short;
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in fact, the head (=H=) and tail (=T=) must /always be touching/
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(diagonally adjacent and even overlapping both count as touching):
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#+begin_example
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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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#+end_example
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If the head is ever two steps directly up, down, left, or right from the
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tail, the tail must also move one step in that direction so it remains
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close enough:
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#+begin_example
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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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#+end_example
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Otherwise, if the head and tail aren't touching and aren't in the same
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row or column, the tail always moves one step diagonally to keep up:
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#+begin_example
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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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#+end_example
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You just need to work out where the tail goes as the head follows a
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series of motions. Assume the head and the tail both start at the same
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position, overlapping.
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For example:
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#+begin_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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#+end_example
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This series of motions moves the head /right/ four steps, then /up/ four
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steps, then /left/ three steps, then /down/ one step, and so on. After
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each step, you'll need to update the position of the tail if the step
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means the head is no longer adjacent to the tail. Visually, these
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motions occur as follows (=s= marks the starting position as a reference
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point):
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#+begin_example
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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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#+end_example
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After simulating the rope, you can count up all of the positions the
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/tail visited at least once/. In this diagram, =s= again marks the
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starting position (which the tail also visited) and =#= marks other
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positions the tail visited:
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#+begin_example
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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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#+end_example
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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
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positions does the tail of the rope visit at least once?/
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Your puzzle answer was =5619=.
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The first half of this puzzle is complete! It provides one gold star: *
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** --- Part Two ---
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A rope snaps! Suddenly, the river is getting a lot closer than you
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remember. The bridge is still there, but some of the ropes that broke
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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
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choose how to arch your body to avoid being hit. Fortunately, your
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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/
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knots. One knot is still the head of the rope and moves according to the
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series of motions. Each knot further down the rope follows the knot in
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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
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knots marked =H=, =1=, =2=, ..., =9=, the motions now occur as follows:
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#+begin_example
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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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#+end_example
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Now, you need to keep track of the positions the new tail, =9=, visits.
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In this example, the tail never moves, and so it only visits =1=
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position. However, /be careful/: more types of motion are possible than
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before, so you might want to visually compare your simulated rope to the
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one above.
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Here's a larger example:
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#+begin_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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#+end_example
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These motions occur as follows (individual steps are not shown):
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#+begin_example
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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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..........................
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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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..........................
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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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...........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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..........................
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== U 8 ==
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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.........
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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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..........................
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..........................
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== L 8 ==
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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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........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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..........................
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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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..........................
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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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== 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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..........................
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..........................
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..........................
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..........................
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..........................
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...........s.........98765
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.........................4
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.........................3
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.........................2
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.........................1
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.........................H
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== L 25 ==
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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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...........s..............
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..........................
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..........................
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..........................
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..........................
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H123456789................
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== U 20 ==
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H.........................
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1.........................
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2.........................
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3.........................
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4.........................
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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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..........................
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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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#+end_example
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|
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|
Now, the tail (=9=) visits =36= positions (including =s=) at least once:
|
|
|
|
#+begin_example
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|
..........................
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|
..........................
|
|
..........................
|
|
..........................
|
|
..........................
|
|
..........................
|
|
..........................
|
|
..........................
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|
..........................
|
|
#.........................
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|
#.............###.........
|
|
#............#...#........
|
|
.#..........#.....#.......
|
|
..#..........#.....#......
|
|
...#........#.......#.....
|
|
....#......s.........#....
|
|
.....#..............#.....
|
|
......#............#......
|
|
.......#..........#.......
|
|
........#........#........
|
|
.........########.........
|
|
#+end_example
|
|
|
|
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?/
|