3x + 1 semigroup

Okay, let's imagine you have three apples in a basket. You can add one more apple to the basket, right? So now you have four apples. If you add one more apple again, now you have five apples.

That's basically what a 3x + 1 semigroup is. The "3x" means you're starting with a certain number (let's say we start with 4). Then you multiply that number by 3 and add 1. So:

4 x 3 + 1 = 13

Now you take that answer (13) and do it again: multiply by 3 and add 1.

13 x 3 + 1 = 40

And you keep doing that over and over again. It's like you're adding "apples" to the basket each time.

Some numbers will eventually get stuck in a loop, where they keep repeating the same numbers over and over again. But other numbers have been proven to eventually reach a certain point where they'll just keep going up and up and up (which is why it's sometimes called the "hailstone" sequence).

It might seem like a weird thing to do, but mathematicians like to study different patterns and sequences to see what they can learn from them. And the 3x + 1 semigroup is one of the ones they find interesting!