The Puzzling Collatz Conjecture

YouTube // Numberphile
YouTube // Numberphile / YouTube // Numberphile

The Collatz Conjecture is a relatively simple set of math instructions that lead to a puzzling problem. If you run this set of rules on a given number, and repeat the process, where do you end up? In every case that mathematicians have tried since the problem was first posed in 1937, they've ended up at the number 1, but the experts can't prove that this will be the case for all (positive, whole) numbers. Why not?

Here's the sequence: Pick a number that is a positive integer. (For instance, the number 1 or 100 or 10,123,456.) If it's even, divide it by two. If it's odd, multiply it by three and add one. Take the resulting number and keep running the process.

In this video, professor David Eisenbud runs the number 7 through this process and ends up at 1. At present, mathematicians have run all whole numbers up to 2^60 through this process and they all end up at 1. But the tricky bit is that the path back to 1 is often winding and bizarre, not following an obvious pattern. Why? This is genuinely surprising:

If that's not enough for you, here's another six minutes of footage on the same topic:

See also: this highly relevant xkcd comic about the Collatz Conjecture.