1) For small L > 0, Let I(L) be a random maximal collection of disjoint closed intervals each of length L and lying in [0,1] in R. Let |I(L)| denote the total length of all the intervals of I(L). It seems clear that there exists some constant C(1) such that, with probability 1, the limit of |I(L)| as L -> 0 = C. Question: Find C. ----------------- One 2D version of this is: 2) For small L > 0 let D(L) denote a random maximal collection of disjoint closed geometric disks each of diameter L and lying in [0,1]^2 in R^2. Likewise, as L -> 0 what is the limit C(2) of the total area of the disks of D(L) ? ----------------- n) The nD version: In [0,1]^n, what is C(n) for any n ? ----------------- (If you don't like edge effects, these questions can be asked for the cubical n-torus R^n / Z^n instead of [0,1]^n, with the same answer C(n).) --Dan