Pick a positive whole number n; if odd, replace with 3n+1, if even divide by 2, repeat ...

Do exactly that, and keep repeating, to get a nice long sequence of positive whole numbers. For example, if you start with 11, you get: 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, ... Any time anyone has ever done this, the sequence always eventually settled down into the cycle 4, 2, 1, 4, 2, 1, 4, 2, ... Does this really always happen, no matter what positive whole number you start with? The conjecture that, yes, this really does always happen, is known as the Collatz conjecture. The problem is to either prove that it will always happen (just trying a few million examples isn't a proof), or to find an example where this doesn't happen.