Advent of Code 2021 day 6

    Day 6 was the first of the puzzles where the challenge was dealing with large numbers. It even says in the puzzle description that it's modelling exponential growth, with a doubling time of around 8 days.

    Part 1

    For 80 days, its enough to model the population directly as a list of the fish ages. That's the same data structure as the input.

    At each generation, two things happen. All the fish age, dropping their age by 1 (unless their age is 0, when it becomes 6 again). All the fish of age 0 produce a new fish of age 8. The new generation is formed by combing these two processes.

    generation shoal = (age shoal) ++ (birth shoal)
    age = map ageFish
    ageFish 0 = 6
    ageFish n = n - 1
    birth = map (const 8) . filter (== 0)

    I iterate the generation function to produce an infinite list of fish populations, then take the 80th day and count the fish.

    part1 fish = length $ (iterate generation fish)!!80

    Part 2

    Part 1 generates about 10⁵ fish, so that explicit approach is fine. Part 2 ends up generating about 10¹² fish, so we need another approach.

    Rather than tracking each fish individually, I track groups of fish. The example in the puzzle description (after 18 days) can be described as in this table.

    Age Number of fish
    0 3
    1 5
    2 3
    3 2
    4 2
    5 1
    6 5
    7 1
    8 4

    Importantly, this table will only have at most 8 entries, as there are only 8 possible ages of fish. Generating a new generation is just creating a new 8-entry table.

    I call this sort of table a Shoal, and use the convenient IntMap to store it. I build a shoal from a list of fish by sorting the list, grouping like elements, then converting them into the assoc list for M.fromList.

    Ageing a shoal has a wrinke, as both fish of age 0 and age 7 become fish of age 6. Birthing new fish is just finding the number of existing age 0 fish.

    generation :: Shoal -> Shoal
    generation shoal = M.union (age shoal) (birth shoal)
    age :: Shoal -> Shoal
    age shoal = M.insert 6 (was0 + was7) shoal'
      where shoal' = M.mapKeys ageFish shoal
            was0 = M.findWithDefault 0 0 shoal
            was7 = M.findWithDefault 0 7 shoal
    birth :: Shoal -> Shoal
    birth shoal = M.singleton 8 parents
      where parents = M.findWithDefault 0 0 shoal
    mkShoal :: [Int] -> Shoal
    mkShoal = M.fromList 
      . map (\g -> (head g, fromIntegral $ length g)) 
      . group 
      . sort

    Finding the population at a given generation is the same as before: iterate to find all generations, !! to pick out the one we want.


    You can get the code from my locally-hosed Git repo, or from Gitlab.

    Neil Smith

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