Math Game HELP: Solve Hard Task with Invariants

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Discussion Overview

The discussion revolves around a mathematical game involving a positive whole number and a set of transformation rules based on whether the number is even or odd. Participants explore whether it is possible to reach the number 1 after a finite number of steps, considering the connection to invariants and the implications of choosing between two transformations for odd numbers.

Discussion Character

  • Exploratory, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant describes the game and seeks help, suggesting a connection to invariants.
  • Another participant identifies the problem as a variation of the Collatz conjecture, noting its unsolved status.
  • A later reply questions whether the choice between $$3n+1$$ and $$3n-1$$ affects the outcome of the game.
  • One participant proposes a strategy for reducing the number to 1 by selecting the multiple of 4 when dealing with odd numbers, arguing that this will lead to a strictly smaller number in subsequent steps.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the game can always lead to the number 1, and there are competing views regarding the influence of the choice between the two transformations for odd numbers.

Contextual Notes

The discussion does not resolve the implications of the variations in the game or the assumptions underlying the proposed strategies.

Who May Find This Useful

Readers interested in mathematical games, conjectures, and number theory may find this discussion relevant.

Mathick
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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.
 
Last edited:
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Deveno said:
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?
 
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.
 

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