TL;DR

The Advent of Code puzzle 15 from 2016 has more Chinese Remainder Theorem!

As you might have noticed, I’ve been taking a look at the 2016 edition of the puzzles in Advent of Code.

No, this will not be another series!

It so happens that puzzle 15 has a long and involved description about capsules, discs, alignments, exact timings… I had to read it twice and put my brain in full imaginative mode.

Anyway, as I read through it, I started suspecting that it had to do with the Chinese Remainder Theorem. Which, indeed, it does.

First thought was: again?!? Then I had that Back to the Future moment when I realized that this puzzle came before the ones I discussed last December 🙄

Second thought was: I don’t want to understand this problem, I want to COOOODE!.

# He who its first…

… hits twice, right?

So I went ahead, took the relevant functions from cglib’s Numbers.pm, put some parsing, a bit of this, a bit of that… and ended up with the following:

#!/usr/bin/env perl
use 5.024;
use warnings;
use English qw< -no_match_vars >;
use autodie;
use experimental qw< postderef signatures >;
no warnings qw< experimental::postderef experimental::signatures >;
use File::Basename qw< basename >;
use Data::Dumper; $Data::Dumper::Indent = 1; use Storable 'dclone';$|++;

my @stuff;

my $filename = shift || basename(__FILE__) =~ s{\.pl\z}{.tmp}rmxs; open my$fh, '<', $filename; while (<$fh>) {
my ($delay,$n, $position) = m{ \A Disc \s+ \#(\d+) \s+ has \s+ (\d+) \s+ positions .*? at \s+ position \s+ (\d+) }mxs or die$_;
push @stuff, $n, ($delay + $position) %$n;
}
close $fh; say((chinese_remainder_theorem(@stuff))); # chinese_remainder_theorem and egcd below... nothing new  I have to admit that I was a bit unsure about the ($delay + $position) %$n - it was somehow a shot in the dark.

Anyway, I run it over the example input and presto! - it worked! Right off the bat!

$cat 15.tmp Disc #1 has 5 positions; at time=0, it is at position 4. Disc #2 has 2 positions; at time=0, it is at position 1.$ perl 15-1.pl 15.tmp
5


OK, on with my puzzle input then:

$cat 15.input Disc #1 has 13 positions; at time=0, it is at position 1. Disc #2 has 19 positions; at time=0, it is at position 10. Disc #3 has 3 positions; at time=0, it is at position 2. Disc #4 has 7 positions; at time=0, it is at position 1. Disc #5 has 5 positions; at time=0, it is at position 3. Disc #6 has 17 positions; at time=0, it is at position 5.$ perl 15-1.pl 15.input
64118


Only that… NO, it does not work!!!

I hit first… but I hit wrong!

# Back to the paper

This must be my humbling year, because I’m reminded so many times of how many ways I have to fail!

Well, I meant learn 👨‍🎓

Let’s see… if we start at time $T$, the first disk is reached after a delay of $1$ at time $T + 1$, and if its starting position (at $t = 0$) is $P_{1, 0}$ then its position at $T + 1$ will be $T + 1 + p_1 \pmod {n_1}$. We have similar relations for the other discs:

$P_{1, T + 1} \equiv T + 1 + P_{1, 0} \pmod {n_1} \\ P_{2, T + 2} \equiv T + 2 + P_{2, 0} \pmod {n_2} \\ ... \\ P_{i, T + i} \equiv T + i + P_{i, 0} \pmod {n_i}$

If we really need that capsule, each of the left-hand sides MUST be $0$, which brings us to:

$T \equiv -1 - P_{1, 0} \pmod {n_1} \\ T \equiv -2 - P_{2, 0} \pmod {n_2} \\ ... \\ T \equiv -i - P_{i, 0} \pmod {n_i}$

This can also be rewritten as:

$r_1 = T \pmod {n_1} = n_1 - (1 + P_{1, 0} \pmod {n_1}) \\ r_2 = T \pmod {n_2} = n_2 - (2 + P_{2, 0} \pmod {n_2}) \\ ... \\ r_i = T \pmod {n_i} = n_i - (i + P_{i, 0} \pmod {n_i})$

I knew it!

It’s also funny that it worked for the example input: simply put, $1 \equiv -1 \pmod 2$, so the sign flip didn’t matter!

So the right code is actually this… a small change for a program, but a big step ahead for a puzzle solver!

#!/usr/bin/env perl
use 5.024;
use warnings;
use English qw< -no_match_vars >;
use autodie;
use experimental qw< postderef signatures >;
no warnings qw< experimental::postderef experimental::signatures >;
use File::Basename qw< basename >;
use Data::Dumper; $Data::Dumper::Indent = 1; use Storable 'dclone';$|++;

my @stuff;

my $filename = shift || basename(__FILE__) =~ s{\.pl\z}{.tmp}rmxs; open my$fh, '<', $filename; while (<$fh>) {
my ($delay,$n, $position) = m{ \A Disc \s+ \#(\d+) \s+ has \s+ (\d+) \s+ positions .*? at \s+ position \s+ (\d+) }mxs or die$_;
push @stuff, $n,$n - ($delay +$position) % $n; } close$fh;

say((chinese_remainder_theorem(@stuff)));

# ...


Let’s run it…

\$ perl 15.pl 15.input
376777


Yay, this is correct now!

# A final thought

I work in the telecommunications industry and most of my… coding occasions come in relation to relatively small integrations, so I definitely have a biased view.

This said… I wonder if the Chineses discovered this theorem just for the fun of puzzle builders and solvers in the 21st century!

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