Advent of Code

--- Day 24: It Hangs in the Balance ---

   It's Christmas Eve, and Santa is loading up the sleigh for this year's
   deliveries. However, there's one small problem: he can't get the sleigh to
   balance. If it isn't balanced, he can't defy physics, and nobody gets
   presents this year.

   No pressure.

   Santa has provided you a list of the weights of every package he needs to
   fit on the sleigh. The packages need to be split into three groups of
   exactly the same weight, and every package has to fit. The first group goes
   in the passenger compartment of the sleigh, and the second and third go in
   containers on either side. Only when all three groups weigh exactly the same
   amount will the sleigh be able to fly. Defying physics has rules, you know!

   Of course, that's not the only problem. The first group - the one going in
   the passenger compartment - needs as few packages as possible so that Santa
   has some legroom left over. It doesn't matter how many packages are in
   either of the other two groups, so long as all of the groups weigh the same.

   Furthermore, Santa tells you, if there are multiple ways to arrange the
   packages such that the fewest possible are in the first group, you need to
   choose the way where the first group has the smallest quantum entanglement
   to reduce the chance of any "complications". The quantum entanglement of a
   group of packages is the product of their weights, that is, the value you
   get when you multiply their weights together. Only consider quantum
   entanglement if the first group has the fewest possible number of packages
   in it and all groups weigh the same amount.

   For example, suppose you have ten packages with weights 1 through 5 and 7
   through 11. For this situation, some of the unique first groups, their
   quantum entanglements, and a way to divide the remaining packages are as
   follows:

 Group 1;             Group 2; Group 3
 11 9       (QE= 99); 10 8 2;  7 5 4 3 1
 10 9 1     (QE= 90); 11 7 2;  8 5 4 3
 10 8 2     (QE=160); 11 9;    7 5 4 3 1
 10 7 3     (QE=210); 11 9;    8 5 4 2 1
 10 5 4 1   (QE=200); 11 9;    8 7 3 2
 10 5 3 2   (QE=300); 11 9;    8 7 4 1
 10 4 3 2 1 (QE=240); 11 9;    8 7 5
 9 8 3      (QE=216); 11 7 2;  10 5 4 1
 9 7 4      (QE=252); 11 8 1;  10 5 3 2
 9 5 4 2    (QE=360); 11 8 1;  10 7 3
 8 7 5      (QE=280); 11 9;    10 4 3 2 1
 8 5 4 3    (QE=480); 11 9;    10 7 2 1
 7 5 4 3 1  (QE=420); 11 9;    10 8 2

   Of these, although 10 9 1 has the smallest quantum entanglement (90), the
   configuration with only two packages, 11 9, in the passenger compartment
   gives Santa the most legroom and wins. In this situation, the quantum
   entanglement for the ideal configuration is therefore 99. Had there been two
   configurations with only two packages in the first group, the one with the
   smaller quantum entanglement would be chosen.

   What is the quantum entanglement of the first group of packages in the ideal
   configuration?

   Your puzzle answer was 10439961859.

--- Part Two ---

   That's weird... the sleigh still isn't balancing.

   "Ho ho ho", Santa muses to himself. "I forgot the trunk".

   Balance the sleigh again, but this time, separate the packages into four
   groups instead of three. The other constraints still apply.

   Given the example packages above, this would be some of the new unique first
   groups, their quantum entanglements, and one way to divide the remaining
   packages:

 11 4    (QE=44); 10 5;   9 3 2 1; 8 7
 10 5    (QE=50); 11 4;   9 3 2 1; 8 7
 9 5 1   (QE=45); 11 4;   10 3 2;  8 7
 9 4 2   (QE=72); 11 3 1; 10 5;    8 7
 9 3 2 1 (QE=54); 11 4;   10 5;    8 7
 8 7     (QE=56); 11 4;   10 5;    9 3 2 1

   Of these, there are three arrangements that put the minimum (two) number of
   packages in the first group: 11 4, 10 5, and 8 7. Of these, 11 4 has the
   lowest quantum entanglement, and so it is selected.

   Now, what is the quantum entanglement of the first group of packages in the
   ideal configuration?

   Your puzzle answer was 72050269.

   Both parts of this puzzle are complete! They provide two gold stars: **

   At this point, all that is left is for you to admire your advent calendar.

   If you still want to see it, you can get your puzzle input.

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