Timothy Lochner

I really enjoy Numberphile videos. This one has be a bit perplexed -- not by the problem itself, but rather the intoned mystery around it. When expressed with binary digits, a clear pattern emerges. https://youtu.be/5mFpVDpKX70 

Here's some properties explaining why you always end up at 1. A) 1 left shifted by multiples of two are always one greater than a multiple of three. 3, 15, 63, 255, etc. B) The x/2 step just eats 0s on the right. (ie, right shift the 0s away)

C) The +1 step converts groups of 1s on the right into groups of 0s on the right (which get eaten by x/2). D) x3 can also be though of as x2+x, which is a left shift of x + x. You can think of this as a way of filling "middle" zeros with 1s.