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Thread: jumping flea in discrete directions

  1. #1
    Nov 2010

    jumping flea in discrete directions

    I have got this problem.
    we have an infinite plane covered by unit squares each one carrying an arrow directed in one of the fourth directions (N, S, W, E). a flea is standing in one of these squares and jump in the next square following th direction of the arrow of the square in which the flea standed before the junp. Right after the jump the arrow turns 90 degrees in clockwise direction.
    is it possible that the distance of the flea from the starting square is limited?

    for me the answer is no.
    the question is equivalent to say that the path is bounded.
    I tried to prove by induction on the number of squares composing the possible path.
    for n=2 it is straightforward to prove that the flea exit the set of 2 squares (easy also for n=3).
    let's suppose true that for n there is not bounded path.
    add then to this n-squared path a futher square to have a set of n+1 squares. on this n+1 square we have the arrow oriented in such way that when the flea goes in the first time , the flea does not escape the path.
    now, if the never goes on this square, it goes out (for inductive hypothesis). if it goes on it , the flea is rebounded into the n-path and then either the flea exits from other squares or it returns. but the arrow on the n+1 square has rotated and in the end the flea exit the n+1 path.

    what do you think? in your opinion is this proof correct?
    thanks inadvance for your feedback
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  2. #2
    MHF Contributor
    Sep 2012

    Re: jumping flea in discrete directions

    Hey pincopallino.

    There is a theorem in 2D random walks that there is a certain probability that given that something starts at a position (x,y) it will always end up back at that position in some finite future time.

    This should give you an idea of distance, but if you wanted to speak about specific intervals of time then this is a bit different.

    You should be able to dig up these results by searching google for limit theorems of 2D random walks.
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