MHB 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 $$n$$ (written on the board) and we write a new one. If number $$n$$ is even, then we write number $$\frac{n}{2}$$ on the board. If number $$n$$ is odd, then we choose one of the numbers: $$3n-1$$ or $$3n+1$$ 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 $$3n+1$$ or $$3n-1$$. 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.