140 lines
5.4 KiB
Org Mode
140 lines
5.4 KiB
Org Mode
** --- Day 15: Beacon Exclusion Zone ---
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You feel the ground rumble again as the distress signal leads you to a
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large network of subterranean tunnels. You don't have time to search
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them all, but you don't need to: your pack contains a set of deployable
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/sensors/ that you imagine were originally built to locate lost Elves.
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The sensors aren't very powerful, but that's okay; your handheld device
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indicates that you're close enough to the source of the distress signal
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to use them. You pull the emergency sensor system out of your pack, hit
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the big button on top, and the sensors zoom off down the tunnels.
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Once a sensor finds a spot it thinks will give it a good reading, it
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attaches itself to a hard surface and begins monitoring for the nearest
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signal source /beacon/. Sensors and beacons always exist at integer
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coordinates. Each sensor knows its own position and can /determine the
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position of a beacon precisely/; however, sensors can only lock on to
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the one beacon /closest to the sensor/ as measured by the
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[[https://en.wikipedia.org/wiki/Taxicab_geometry][Manhattan distance]].
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(There is never a tie where two beacons are the same distance to a
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sensor.)
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It doesn't take long for the sensors to report back their positions and
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closest beacons (your puzzle input). For example:
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#+begin_example
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Sensor at x=2, y=18: closest beacon is at x=-2, y=15
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Sensor at x=9, y=16: closest beacon is at x=10, y=16
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Sensor at x=13, y=2: closest beacon is at x=15, y=3
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Sensor at x=12, y=14: closest beacon is at x=10, y=16
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Sensor at x=10, y=20: closest beacon is at x=10, y=16
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Sensor at x=14, y=17: closest beacon is at x=10, y=16
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Sensor at x=8, y=7: closest beacon is at x=2, y=10
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Sensor at x=2, y=0: closest beacon is at x=2, y=10
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Sensor at x=0, y=11: closest beacon is at x=2, y=10
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Sensor at x=20, y=14: closest beacon is at x=25, y=17
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Sensor at x=17, y=20: closest beacon is at x=21, y=22
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Sensor at x=16, y=7: closest beacon is at x=15, y=3
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Sensor at x=14, y=3: closest beacon is at x=15, y=3
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Sensor at x=20, y=1: closest beacon is at x=15, y=3
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#+end_example
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So, consider the sensor at =2,18=; the closest beacon to it is at
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=-2,15=. For the sensor at =9,16=, the closest beacon to it is at
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=10,16=.
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Drawing sensors as =S= and beacons as =B=, the above arrangement of
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sensors and beacons looks like this:
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#+begin_example
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1 1 2 2
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0 5 0 5 0 5
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0 ....S.......................
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1 ......................S.....
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2 ...............S............
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3 ................SB..........
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4 ............................
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5 ............................
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6 ............................
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7 ..........S.......S.........
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8 ............................
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9 ............................
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10 ....B.......................
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11 ..S.........................
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12 ............................
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13 ............................
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14 ..............S.......S.....
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15 B...........................
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16 ...........SB...............
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17 ................S..........B
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18 ....S.......................
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19 ............................
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20 ............S......S........
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21 ............................
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22 .......................B....
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#+end_example
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This isn't necessarily a comprehensive map of all beacons in the area,
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though. Because each sensor only identifies its closest beacon, if a
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sensor detects a beacon, you know there are no other beacons that close
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or closer to that sensor. There could still be beacons that just happen
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to not be the closest beacon to any sensor. Consider the sensor at
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=8,7=:
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#+begin_example
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1 1 2 2
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0 5 0 5 0 5
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-2 ..........#.................
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-1 .........###................
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0 ....S...#####...............
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1 .......#######........S.....
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2 ......#########S............
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3 .....###########SB..........
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4 ....#############...........
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5 ...###############..........
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6 ..#################.........
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7 .#########S#######S#........
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8 ..#################.........
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9 ...###############..........
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10 ....B############...........
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11 ..S..###########............
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12 ......#########.............
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13 .......#######..............
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14 ........#####.S.......S.....
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15 B........###................
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16 ..........#SB...............
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17 ................S..........B
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18 ....S.......................
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19 ............................
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20 ............S......S........
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21 ............................
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22 .......................B....
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#+end_example
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This sensor's closest beacon is at =2,10=, and so you know there are no
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beacons that close or closer (in any positions marked =#=).
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None of the detected beacons seem to be producing the distress signal,
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so you'll need to work out where the distress beacon is by working out
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where it /isn't/. For now, keep things simple by counting the positions
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where a beacon cannot possibly be along just a single row.
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So, suppose you have an arrangement of beacons and sensors like in the
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example above and, just in the row where =y=10=, you'd like to count the
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number of positions a beacon cannot possibly exist. The coverage from
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all sensors near that row looks like this:
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#+begin_example
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1 1 2 2
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0 5 0 5 0 5
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9 ...#########################...
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10 ..####B######################..
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11 .###S#############.###########.
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#+end_example
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In this example, in the row where =y=10=, there are =26= positions where
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a beacon cannot be present.
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Consult the report from the sensors you just deployed. /In the row where
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=y=2000000=, how many positions cannot contain a beacon?/
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