Math Game HELP: Solve Hard Task with Invariants

[Mathick]

HELP - math game

A positive whole number was written on the board. In each step we rub out the number \(\displaystyle n\) (written on the board) and we write a new one. If number \(\displaystyle n\) is even, then we write number \(\displaystyle \frac{n}{2}\) on the board. If number \(\displaystyle n\) is odd, then we choose one of the numbers: \(\displaystyle 3n-1\) or \(\displaystyle 3n+1\) and we write it down on the board. Decide, if after finite amount of steps, we can obtain the number 1 one the board (no matter which number was written on the board at the beginning).

Please help. I can't figure it out. I know that it can be connected to invariants.

 

[Deveno]

This is the Collatz conjecture. It is currently unsolved.

 

[Mathick]

This is the Collatz conjecture. It is currently unsolved.

Yes, I found it but there is one extra variation. You can choose either \(\displaystyle 3n+1\) or \(\displaystyle 3n-1\). Doesn't it have an influence on the result?

 

[Opalg]

If at some stage of the game you have an odd number $n$ then one of the numbers $3n\pm1$ will be a multiple of $4$. Choose that one. Then the next two steps will take you to the number $\dfrac{3n\pm1}4$, which is strictly smaller than $n$. This process of making numbers strictly smaller will inevitably bring you down to $1$ eventually.