Main post link -> http://en.wikipedia.org/wiki/Collatz_conjecture
Start with any natural number. If it is even, divide by two, if odd, multiply by three and add one. Repeat the same method on the new number, and the one after that, so on. Eventually you should reach one, no matter what number you started with. eg. 6\(\implies\)3 \(\implies\)10 \(\implies\)5 \(\implies\)16 \(\implies\)8 \(\implies\)4 \(\implies\)2 \(\implies\)1
If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence which does not contain 1. Such a sequence might enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found.