Terence Tao

So you take any natural number, like, say, 13, and you apply the following procedure to it.

So if it's even, you divide it by 2.

And if it's odd, you multiply it by 3 and add 1.

So even numbers get smaller, odd numbers get bigger.

So 13 will become 40, because 13 times 3 is 39, add 1, you get 40.

So it's a simple process.

For odd numbers and even numbers, they're both very easy operations.

And then you put them together, it's still reasonably simple.

But then you ask what happens when you iterate it.

You take the output that you just got and feed it back in.

So 13 becomes 40.

40 is now even, divided by 2 is 20.

20 is still even, divided by 2 is 10.

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.