Simple Math Problem No One Solved - Collatz Conjecture
Simple Math Problem No One Solved - Collatz Conjecture
Pick any positive integer. If it is even, divide it by 2. If it is odd, multiply it by 3 and add 1. The Collatz conjecture says this process always eventually reaches 1.
This has not been proven. It is still a conjecture: no one has shown that every positive integer must reach 1, and no counterexample is known.
The strange part is how differently small starting numbers behave before they fall back down.
Another way to look at the conjecture is to ask: for each starting number, how high does it climb before it falls to 1?
The usual conjecture is only about positive integers. If you apply the same rule to negative integers, the picture changes: there are multiple disconnected loops.
To understand why the sequence tends to fall statistically, ignore the even values for a moment and look only at jumps from one odd number to the next odd number.
The leading digits inside many Collatz sequences settle toward a familiar distribution: Benford's law.
The same rule can also be drawn as a path: turn anticlockwise if odd, clockwise if even, and move one step at a time.