The Collatz conjecture.
Define $f(n) = n/2$ if $n$ is even, and $3n+1$ if $n$ is odd.
(Parenthetical silly question: what do we see if we make $n/2$ if $n$ is odd and $3n+1$ if $n$ is even? Need to say what to do if $n$ is not an integer)
The Collatz conjecture says that for all $n$, for sufficiently large $k$, $f^k(n) \in \{1,2,4\}$.
We can track how many iterations it takes to reach 1.
Define $h(n)$, the hailstone function, to be the number of iterations it takes for $n$ to reach 1.
Compute a list of values $[n,h(n)]$.
Plot this for various ranges of $n$.

Finding champions.
A number $n$ is a champion if $h(n)>h(k)$ for all $k<n$, that is, if it has the largest $h$ value we've seen so far.
So, for example, 27 is champion.
Write code to do the following:
Set up an empty list of champions
Set the current maxh to 1
For each $n$, if $h(n)> $ maxh, replace maxh by $h(n)$, and append $n$ to the list of champions.
