# 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.

## Code

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