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I was watching the following video on Random Walks: https://www.youtube.com/watch?v=iH2kATv49rc

Here, the following point is discussed:

  • Suppose you have an infinite square grid.
  • In each "step", you can move one "step" in the Up, Down, Left or Right Direction
  • Suppose you are at the origin at step = 0
  • Given this information, is only possible to return to the origin in an even number of steps

My Question: While I think this is obvious, I was interested in knowing how statements like this can be mathematically proven.

I think this can be proven using mathematical induction ($n$ is the number of steps).

  • When $n$ = 0, you are at the origin
  • Assume that in $2n$ steps you can return to the origin: And $2n$ is an even number
  • Now consider $2n+2$ steps : $2n+2$ is also an even number
  • Since you can only return in an even number of steps and $2n+2$ is an even number - the proof is concluded.

Is this sufficient to conclude the proof?

Thanks!

References:

stats_noob
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    No, it is not a valid proof. It nowhere treats the case of treks of odd length. For instance, a purported round trip of length 9 steps is not invalidated anywhere in your logic. – Paul Tanenbaum Nov 19 '23 at 05:08
  • @Paul Tanenbaum: thank you so much for your reply! Can you please suggest a different way to prove this if you have time? thank you so much! – stats_noob Nov 19 '23 at 17:49

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