A visualisation of the intricacies of the Collatz Conjecture for the first 1000 integers:
Start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.
In this particular animation, the virtual 'path' an integer must take via the above rules will take a left-hand turn if the decision is to halve the number, and a right-hand turn if the decision is to multiply by 3 and add 1. This repeats until the result reaches 1. The animation shows some categorisation for the first 1000 integers to examine the differences in the length (how many moves to get to 1) and direction (types of move to get to 1) of their 'paths':
Prime numbers, Square numbers, Triangular numbers, and others.