Mathematics
The Collatz Conjecture and Its Computational Exploration
Quick fact
The Collatz conjecture has been verified for every starting number up to 2^68 (about 295 billion billion) without a single counterexample, yet no one has been able to prove it holds for all positive integers.
Why this is interesting
Start with any positive integer. If it's even, halve it; if it's odd, triple it and add one. No matter where you start, you always end up at 1—or do you?