You find yourselves on the roof of a top-secret Easter Bunny installation.
While The Historians do their thing, you take a look at the familiar huge antenna. Much to your surprise, it seems to have been reconfigured to emit a signal that makes people 0.1% more likely to buy Easter Bunny brand Imitation Mediocre Chocolate as a Christmas gift! Unthinkable!
Scanning across the city, you find that there are actually many such antennas. Each antenna is tuned to a specific frequency indicated by a single lowercase letter, uppercase letter, or digit. You create a map (your puzzle input) of these antennas. For example:
............
........0...
.....0......
.......0....
....0.......
......A.....
............
............
........A...
.........A..
............
............
The signal only applies its nefarious effect at specific antinodes based on the resonant frequencies of the antennas. In particular, an antinode occurs at any point that is perfectly in line with two antennas of the same frequency - but only when one of the antennas is twice as far away as the other. This means that for any pair of antennas with the same frequency, there are two antinodes, one on either side of them.
So, for these two antennas with frequency a, they create the two antinodes marked with #:
..........
...#......
..........
....a.....
..........
.....a....
..........
......#...
..........
..........
Adding a third antenna with the same frequency creates several more antinodes. It would ideally add four antinodes, but two are off the right side of the map, so instead it adds only two:
..........
...#......
#.........
....a.....
........a.
.....a....
..#.......
......#...
..........
..........
Antennas with different frequencies don't create antinodes; A and a count as different frequencies. However, antinodes can occur at locations that contain antennas. In this diagram, the lone antenna with frequency capital A creates no antinodes but has a lowercase-a-frequency antinode at its location:
..........
...#......
#.........
....a.....
........a.
.....a....
..#.......
......A...
..........
..........
The first example has antennas with two different frequencies, so the antinodes they create look like this, plus an antinode overlapping the topmost A-frequency antenna:
......#....#
...#....0...
....#0....#.
..#....0....
....0....#..
.#....A.....
...#........
#......#....
........A...
.........A..
..........#.
..........#.
Because the topmost A-frequency antenna overlaps with a 0-frequency antinode, there are 14 total unique locations that contain an antinode within the bounds of the map.
Calculate the impact of the signal. How many unique locations within the bounds of the map contain an antinode?
Your puzzle answer was 299.
Watching over your shoulder as you work, one of The Historians asks if you took the effects of resonant harmonics into your calculations.
Whoops!
After updating your model, it turns out that an antinode occurs at any grid position exactly in line with at least two antennas of the same frequency, regardless of distance. This means that some of the new antinodes will occur at the position of each antenna (unless that antenna is the only one of its frequency).
So, these three T-frequency antennas now create many antinodes:
T....#....
...T......
.T....#...
.........#
..#.......
..........
...#......
..........
....#.....
..........
In fact, the three T-frequency antennas are all exactly in line with two antennas, so they are all also antinodes! This brings the total number of antinodes in the above example to 9.
The original example now has 34 antinodes, including the antinodes that appear on every antenna:
##....#....#
.#.#....0...
..#.#0....#.
..##...0....
....0....#..
.#...#A....#
...#..#.....
#....#.#....
..#.....A...
....#....A..
.#........#.
...#......##
Calculate the impact of the signal using this updated model. How many unique locations within the bounds of the map contain an antinode?
Your puzzle answer was 1032.
A very complicated task description that ends up in a relatively unspectacular implementation. In part 1, some care has to be taken to (a) only count each antenna pair once, (b) not automatically count the antennas themselves as antinodes, unless they are referenced by some other antenna pair, and (c) to precisely exclude results outside the valid area. Even though contraint (c) requires individual component access, complex numbers prove as useful tools here once again.
Part 2 is a great surprise, because it's actually simpler than part 1. It's easy to miss in the description (trust me on that one!), but the antennas now are antinodes themselves, so you just need to splat antinodes at each antenna and any multiple of the distance from that to any other same-frequency antenna. You could compute how many antinodes to put, but you could also rightly assume that it's never more than the map size and prune out-of-map antinodes during the final count, as was a good idea for part 1 already.
The interesting thing is, as a friend pointed out, that the simplicity of part 2 is only due to the construction of the input data. The X and Y deltas of any same-frequency antenna pair are always coprime, meaning that "exactly in line with two antennas" really only yields multiples of the X and Y deltas. If delta-X and delta-Y were both even, for example, there would be additional antinodes at the halfway point between them - but that just never happens in the input data, and at around 500 antenna pairs for a typical input, that can't be coincidence.
- Part 1, Python: 247 bytes, <100 ms
- Part 2, Python: 233 bytes, <100 ms