Simple unsolved math problem, 1

In 1937 Lothar Collatz proposed the 3n+1 conjecture (known by a long list of aliases), is stated as follows.

First, we define the function $f$ on the set of positive integers:

If the number $n$ is even, divide it by two: $f(n)=n/2$.
If the number $n$ is odd, triple it and add one: $f(n)=3n+1$.

In modular arithmetic notation, define the function $f$ as follows:
$f(n)= {n/2},\ if \ n\equiv 0 \pmod 2$, and $f(n)= {3n+1},\ if \ n\equiv 1 \pmod 2$. Believe it or not, this is the restriction to the positive integers of the complex-valued map $(2+7z-(2+5z)\cos(\pi z))/4$.

The 3n+1 conjecture is: The sequence
$n,\ f(n),\ f^2(n)=f(f(n)),\ f^3(n)=f(f^2(n)),\ \dots$
will eventually reach the number 1, regardless of which positive integer $n$ is chosen initially.

This is still unsolved, though a lot of people have worked on it. For a recent survey of results, see the paper by Chamberland.