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Math Help - Dividing a square with different parallel lines and minimize distance between.

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    Dividing a square with different parallel lines and minimize distance between.

    A square of side 1 is divided into "a" strips by "a-1" equally spaced
    red lines parallel to a side, and into "b" strips by "b-1" equally spaced
    blue lines parallel to the red lines. Suppose that a does not divide b
    and that b does not divide a. What is the smallest possible distance
    between a red line and a blue line?


    Edit: Actually, I think case one may be that assume (a,b) =/= 1. In that case since a does not divide b and b does not divide a, there must be at least one c such that (a,b) = c. In this case the red and blue lines most coincide, and so the minimum distance is 0. What about the case when (a,b) = 1?
    Last edited by libzdolce; October 18th 2011 at 11:00 PM.
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  2. #2
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    Re: Dividing a square with different parallel lines and minimize distance between.

    Quote Originally Posted by libzdolce View Post
    A square of side 1 is divided into "a" strips by "a-1" equally spaced
    red lines parallel to a side, and into "b" strips by "b-1" equally spaced
    blue lines parallel to the red lines. Suppose that a does not divide b
    and that b does not divide a. What is the smallest possible distance
    between a red line and a blue line?


    Edit: Actually, I think case one may be that assume (a,b) =/= 1. In that case since a does not divide b and b does not divide a, there must be at least one c such that (a,b) = c. In this case the red and blue lines most coincide, and so the minimum distance is 0. What about the case when (a,b) = 1?
    I assume you mean that the strips must be of equal width, so that the red lines are at distances k/a\ (1\leqslant k\leqslant a-1) from one side of the square, and similarly for the blue lines. It then looks as though the minimum distance from a red line to a blue line should be 1/(ab).

    Reason: given that (a,b) = 1, there exist integers p, q such that |pa-qb| = 1. Then \left|\tfrac pb - \tfrac qa\right| = \tfrac1{ab}.
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