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19 - Advent of Code
2021</title>
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16 Oh, hello! Funny seeing you here.
18 I appreciate your enthusiasm, but you aren't going to find much down here.
19 There certainly aren't clues to any of the puzzles. The best surprises don't
20 even appear in the source until you unlock them for real.
22 Please be careful with automated requests; I'm not a massive company, and I can
23 only take so much traffic. Please be considerate so that everyone gets to play.
25 If you're curious about how Advent of Code works, it's running on some custom
26 Perl code. Other than a few integrations (auth, analytics, social media), I
27 built the whole thing myself, including the design, animations, prose, and all
30 The puzzles are most of the work; preparing a new calendar and a new set of
31 puzzles each year takes all of my free time for 4-5 months. A lot of effort
32 went into building this thing - I hope you're enjoying playing it as much as I
33 enjoyed making it for you!
35 If you'd like to hang out, I'm @ericwastl on Twitter.
90 <header><div><h1 class=
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98 <article class=
"day-desc"><h2>--- Day
19: Beacon Scanner ---
</h2><p>As your
<a href=
"17">probe
</a> drifted down through this area, it released an assortment of
<em>beacons
</em> and
<em>scanners
</em> into the water. It's difficult to navigate in the pitch black open waters of the ocean trench, but if you can build a map of the trench using data from the scanners, you should be able to safely reach the bottom.
</p>
99 <p>The beacons and scanners float motionless in the water; they're designed to maintain the same position for long periods of time. Each scanner is capable of detecting all beacons in a large cube centered on the scanner; beacons that are at most
1000 units away from the scanner in each of the three axes (
<code>x
</code>,
<code>y
</code>, and
<code>z
</code>) have their precise position determined relative to the scanner. However, scanners cannot detect other scanners. The submarine has automatically summarized the relative positions of beacons detected by each scanner (your puzzle input).
</p>
100 <p>For example, if a scanner is at
<code>x,y,z
</code> coordinates
<code>500,
0,-
500</code> and there are beacons at
<code>-
500,
1000,-
1500</code> and
<code>1501,
0,-
500</code>, the scanner could report that the first beacon is at
<code>-
1000,
1000,-
1000</code> (relative to the scanner) but would not detect the second beacon at all.
</p>
101 <p>Unfortunately, while each scanner can report the positions of all detected beacons relative to itself,
<em>the scanners do not know their own position
</em>. You'll need to determine the positions of the beacons and scanners yourself.
</p>
102 <p>The scanners and beacons map a single contiguous
3d region. This region can be reconstructed by finding pairs of scanners that have overlapping detection regions such that there are
<em>at least
12 beacons
</em> that both scanners detect within the overlap. By establishing
12 common beacons, you can precisely determine where the scanners are relative to each other, allowing you to reconstruct the beacon map one scanner at a time.
</p>
103 <p>For a moment, consider only two dimensions. Suppose you have the following scanner reports:
</p>
104 <pre><code>--- scanner
0 ---
114 <p>Drawing
<code>x
</code> increasing rightward,
<code>y
</code> increasing upward, scanners as
<code>S
</code>, and beacons as
<code>B
</code>, scanner
<code>0</code> detects this:
</p>
120 <p>Scanner
<code>1</code> detects this:
</p>
125 <p>For this example, assume scanners only need
3 overlapping beacons. Then, the beacons visible to both scanners overlap to produce the following complete map:
</p>
131 <p>Unfortunately, there's a second problem: the scanners also don't know their
<em>rotation or facing direction
</em>. Due to magnetic alignment, each scanner is rotated some integer number of
90-degree turns around all of the
<code>x
</code>,
<code>y
</code>, and
<code>z
</code> axes. That is, one scanner might call a direction positive
<code>x
</code>, while another scanner might call that direction negative
<code>y
</code>. Or, two scanners might agree on which direction is positive
<code>x
</code>, but one scanner might be upside-down from the perspective of the other scanner. In total, each scanner could be in any of
24 different orientations: facing positive or negative
<code>x
</code>,
<code>y
</code>, or
<code>z
</code>, and considering any of four directions
"up" from that facing.
</p>
132 <p>For example, here is an arrangement of beacons as seen from a scanner in the same position but in different orientations:
</p>
133 <pre><code>--- scanner
0 ---
173 <p>By finding pairs of scanners that both see at least
12 of the same beacons, you can assemble the entire map. For example, consider the following report:
</p>
174 <pre><code>--- scanner
0 ---
311 <p>Because all coordinates are relative, in this example, all
"absolute" positions will be expressed relative to scanner
<code>0</code> (using the orientation of scanner
<code>0</code> and as if scanner
<code>0</code> is at coordinates
<code>0,
0,
0</code>).
</p>
312 <p>Scanners
<code>0</code> and
<code>1</code> have overlapping detection cubes; the
12 beacons they both detect (relative to scanner
<code>0</code>) are at the following coordinates:
</p>
313 <pre><code>-
618,-
824,-
621
326 <p>These same
12 beacons (in the same order) but from the perspective of scanner
<code>1</code> are:
</p>
327 <pre><code>686,
422,
578
340 <p>Because of this, scanner
<code>1</code> must be at
<code>68,-
1246,-
43</code> (relative to scanner
<code>0</code>).
</p>
341 <p>Scanner
<code>4</code> overlaps with scanner
<code>1</code>; the
12 beacons they both detect (relative to scanner
<code>0</code>) are:
</p>
342 <pre><code>459,-
707,
401
355 <p>So, scanner
<code>4</code> is at
<code>-
20,-
1133,
1061</code> (relative to scanner
<code>0</code>).
</p>
356 <p>Following this process, scanner
<code>2</code> must be at
<code>1105,-
1205,
1229</code> (relative to scanner
<code>0</code>) and scanner
<code>3</code> must be at
<code>-
92,-
2380,-
20</code> (relative to scanner
<code>0</code>).
</p>
357 <p>The full list of beacons (relative to scanner
<code>0</code>) is:
</p>
358 <pre><code>-
892,
524,
684
438 <p>In total, there are
<code><em>79</em></code> beacons.
</p>
439 <p>Assemble the full map of beacons.
<em>How many beacons are there?
</em></p>
441 <p>Your puzzle answer was
<code>355</code>.
</p><article class=
"day-desc"><h2 id=
"part2">--- Part Two ---
</h2><p>Sometimes, it's a good idea to appreciate just how
<span title=
"The deepest parts of the ocean are about as deep as the altitude of a normal commercial aircraft, roughly 11 kilometers or 36000 feet.">big
</span> the ocean is. Using the
<a href=
"https://en.wikipedia.org/wiki/Taxicab_geometry" target=
"_blank">Manhattan distance
</a>, how far apart do the scanners get?
</p>
442 <p>In the above example, scanners
<code>2</code> (
<code>1105,-
1205,
1229</code>) and
<code>3</code> (
<code>-
92,-
2380,-
20</code>) are the largest Manhattan distance apart. In total, they are
<code>1197 +
1175 +
1249 =
<em>3621</em></code> units apart.
</p>
443 <p><em>What is the largest Manhattan distance between any two scanners?
</em></p>
445 <p>Your puzzle answer was
<code>10842</code>.
</p><p class=
"day-success">Both parts of this puzzle are complete! They provide two gold stars: **
</p>
446 <p>At this point, you should
<a href=
"/2021">return to your Advent calendar
</a> and try another puzzle.
</p>
447 <p>If you still want to see it, you can
<a href=
"19/input" target=
"_blank">get your puzzle input
</a>.
</p>
448 <p>You can also
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"share">[Share
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451 ></span>]
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</p>
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