Take any positive integer n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinitely. Does the sequence eventually reach 1, regardless of the initial value? For instance, if you start with the number 75,128,138,247, you eventually reach 1 after 1228 steps. If you start with the number 27, you climb as high as 9,232, but eventually reach 1 after 41 steps.
This is supposed to be a very difficult problem. Note that if a sequence reaches any power of 2 (say, 64) or any intermediate number found in the trillions of trillions of such sequences that are known to reach 1, then the sequence in question will obviously reach 1 too. For a sequence not to reach 1, the first element (as well as any subsequent element) would have to be different from any initial or intermediate number found in any series identified as reaching 1 so far. This makes it highly unlikely, yet the conjecture has not been proved yet.