Main Concept

The Collatz Conjecture, also known as the $\mathbf{3}\mathit{n}\mathbf{\+}\mathbf{1}$ Conjecture, states that if you choose any natural number and repeatedly perform a "Half Or Triple Plus One" (HOTPO) procedure, you will always eventually reach 1.

HOTPO: Choose an integer $n$ greater than 0. Then:

Now, apply this same process to the resulting number, and keep repeating this process indefinitely. You end up with a sequence of positive integers:

$7 \to 3\times 7\+1\= 22 \to 22÷2 \= 11 \to 3\times 11\+1\=34 \to 34÷2 \= 17 ..\.$

In the resulting sequence, sometimes the numbers go up and sometimes they go down, and it can be hard to predict whether the number even a few iterations away will be higher or lower than the current number. If the conjecture is right though, eventually the number sequence always decreases to 1.

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