Terence Tao

5, and then 5 times 3 plus 1 is 16.

And then 8, 4, 2, 1.

And then from 1 it goes 1, 4, 2, 1, 4, 2, 1.

It cycles forever.

So this sequence I just described, 13, 40, 20, 10, so both, these are also called Hailstone Sequences.

because there's an oversimplified model of hailstorm formation, which is not actually quite correct, but it's somehow taught to high school students as a first approximation, is that a little nugget of ice gets an ice crystal.

It forms in a cloud, and it goes up and down because of the wind, and sometimes when it's cold, it acquires a bit more mass, and maybe it melts a little bit, and this process of going up and down creates this partially melted ice, which eventually becomes this hailstorm, and eventually it falls down to the earth.

So the conjecture is that no matter how high you start up, like you take a number which is in the millions or billions, this process that goes up if you're odd and down if you're even, it eventually goes down to earth all the time.

No matter where you start with this very simple algorithm, you end up at one.

And you might climb for a while, come down.

Yeah, if you plot it, these sequences, they look like Brownian motion.

They look like the stock market.

They just go up and down in a seemingly random pattern.

And in fact, usually that's what happens, that if you plug in a random number, you can actually prove, at least initially, that it would look like a random walk.

And that's actually a random walk with a downward drift.

It's like if you're always gambling on a roulette at the casino with odds slightly weighted against you.

So sometimes you win, sometimes you lose.

But over, in the long run, you lose a bit more than you win.

And so normally your wallet will go to zero if you just keep playing over and over again.

Yes.