I got 1/3, but with a "different" distribution. (I took symmetrized pairs rather than pairs. But I realized after I wrote this comment that folding a square in

I got 1/3, but with a "different" distribution. (I took symmetrized pairs rather than pairs. But I realized after I wrote this comment that folding a square in half preserves relative area so symmetrizing makes no difference.) I started off with a flat torus (two ordered points on the circle) and then quotiented out by (x, y) ~ (y, x) (since the chord corresponding to both is the same) to get a triangle as in the diagram [0]. By geometric intuition/trigonometry/etc it's clear that a chord has length < sqrt(3) iff the difference between the angles x and y is less than 2pi/3. The region corresponding to pairs with distance distance less than 2pi/3 is shaded in [0]. One adds it up and sees that the unshaded region is 1/3 of the whole. [0]: http://i.imgur.com/ow8Dkgd.png http://i.imgur.com/ow8Dkgd.png