The inspiration for this project is The Collatz Conjecture. The wonderful thing about the Collatz Conjecture is that it only requires addition, multiplication, and division and anyone can understand it well enough to play with it. But at the same time it's an unsolved problem in mathematics.

The idea is that you start from any positive integer (1, 2, 3, ...) and apply the following rule: if your number is even, divide by two to get the next one. If it's odd, multiply by 3 and add 1 to get the next. The unproven conjecture is that any number you start from will eventually end up at 1. For example if you start at 12 it goes 6, 3, 10, 5, 16, 8, 4, 2, 1. Try it. As far as anyone can tell it always goes to one. Actually if you keep going what happens is you always end up in the loop 1, 4, 2, 1...

I decided to make a tool that let's you play with this rule. Use whatever integer values you want for dividing, multiplying, and adding, and see what happens. For example you might try multiplying by 2 instead of 3. You'll find that with this rule the numbers always blow up and never converge to 1 or anything else. Or you might try the original problem but with -1 instead of 1. This leads to ending up in many different loops depending on the starting point.

I heard about this problem from these Numberphile videos: one, and two. And here's the obligatory xkcd.