Thread: 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?

Originally Posted by libzdolce

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 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

Reason: given that (a,b) = 1, there exist integers p, q such that Then