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.