- Pick any positive integer
- If it is even, divide by 2
- If it is odd, multiply by 3 and add 1
- repeat these steps until the cycle repeats
If n is even, n/2
If n is odd, 3n+1
Here's the conjecture: Every number (n) will eventually go to 1 using these rules. So far, every number up to 2^60 has gone to 1. 1 is the only number that repeats itself, going from 1, 3*1+1=4, 4/2=2, 2/2=1, and so on forever. It seems like a simple problem to solve, but it isn't. What seems to make this problem so difficult to prove is that there is no application for it and there is no pattern. It falls under a category of "recreational math". Prime numbers and the hunt for them are also considered recreational, but primes are used in encryption these days (although, a 10^64-digit prime is kind of unwieldy for encrypting your passwords).
Because there is no current application for this problem, there is no cash prize for solving it like there is for the Riemann Hypothesis or the P Versus NP problem. The Collatz Conjecture is a curiosity in math and I love the problem, but I know I'll never see a solution for it in my lifetime. For now, all I know is that any power of 2 goes to 1 and doubling any number that goes to one will simply also go to one. Here's a video about this problem:
Play around with yourself if you are mathematically inclined. and if you're really inclined, try to solve the Millennium Prize Problems. Those should be easy </sarcasm>.