Buffon's problem with a star of needles and a lattice of parallelograms

Abstract

A star of n (n greater than or equal to 2) line segments (needles) of equal length with common endpoint and constant angular spacing is randomly placed onto a lattice which is the union of two families of equidistant lines in the plane with angle alpha between the nonparallel lines. For odd n, we calculate the probabilities of exactly i intersections between the star and the lattice (for even n, see [3]). Using a geometrical method, we derive the limit distribution function of the relative number of intersections as n tends to infinity. This function is independent of alpha. We show that the relative numbers for each of the two families are asymptotically independent random variables.