= Collatz Chains in Haskell

== Title header

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The Simplest Math Problem No One Can Solve - Collatz Conjecture
<https://youtu.be/094y1Z2wpJg>

Why do we want to do this?

> collatzStep :: Integer -> Integer
> collatzStep n
>   | even n = n `div` 2
>   | odd  n = 3 * n + 1

With this, we can iterate over it.

> collatzStep' :: Integer -> Integer
> collatzStep' n
>   | n == 1 = error "done"
>   | even n = n `div` 2
>   | odd  n = 3 * n + 1

> result :: [Integer]
> result = iterate collatzStep' 5


collatz collect
generate the collatz sequence and return it

> cc :: Integer -> [Integer]
> cc n = cc' [] n
>   where
>     cc' :: [Integer] -> Integer -> [Integer]
>     cc' acc n
>       | n ==    1  = acc <>   [1]
>       | n ==    0  = acc <>   [0]
>       | n ==  (-1) = acc <>  [-1]
>       | n ==  (-5) = acc <>  [-5]
>       | n == (-17) = acc <> [-17]
>       | even n = cc' acc' (n `div` 2)
>       | odd  n = cc' acc' (3*n + 1)
>       where acc' = acc <> [n]

> a :: [Integer]
> a = reverse $ cc $ 2^1_000+1

> b :: [Integer]
> b = reverse $ cc $ 2^1_000+2

> comparingChains :: [Bool]
> comparingChains = zipWith (==) a b

Now if we collect the chain lengths of large numbers we see something slightly horrifying:

> x = map (length . cc) [2^1_000+10..]

This prints us the list of lengths of the chain as:
[7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7430,7249,7249,7249,7249,7249,7249,7249,7249,7249,7430,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7430,7430,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7430,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7430,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7430,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249,7249...]

λ> cc 13
[13,40,20,10,5,16,8,4,2,1]
λ> cc 12
[12,6,3,10,5,16,8,4,2,1]

Now this makes sense with small numbers.
But I find it weird with large numbers.

> f :: Integer -> Integer
> f n = s 1 n
>   where
>     s :: Integer -> Integer -> Integer
>     s i n
>       | n ==    1  = i
>       | n ==    0  = i
>       | n ==  (-1) = i
>       | n ==  (-5) = i
>       | n == (-17) = i
>       | even n = s (succ i) (n `div` 2)
>       | odd  n = s (succ i) (3 * n + 1)