# Collatz conjecture became a mathematical illustration, why freedom of choice matters

Authoritarian rules without alternatives may not ensure the achievement of desired aim, unlike liberal rules giving people a choice. This idea was proved mathematically in the article published by peer-reviewed online journal Academia Letters.

For any natural number X or odd number N, consider two ways of building a sequence of natural numbers.

The first way is that you have no choice other than to move from 2X to X or from N to 3N+1. For example: from 7 you should move to 22, then to 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, and finally to 1.

The second way is that you are allowed to choose whether to move from 2X to X, or from X to 3X+1, or vice versa. Starting from 7, you can move to 2 and then to 1.

Consider a question: is it true that for any natural number you can eventually reach 1, repeating the move?

In the first case, this question is the famous Collatz conjecture a.k.a. 3x+1 problem, which nobody managed to prove yet. It illustrates that strict deterministic regulation of behavior leads to unpredictable results.

In the second case, a positive answer to the question was proved in the article “Autonomous Version of Collatz Conjecture: Freedom of Choice Makes Unsolved Problem Solvable,” published in the online open-access journal Academia Letters.

“The counterintuitive example of making certain outcomes of formal rules with uncertain outcomes via introducing freedom of choice shows heuristic and educational value of the multidisciplinary approach, linking mathematics, philosophy, and legal theory,” wrote Yurii Sheliazhenko, author of the article.