On Buffon's Needle Problem Using Concentric Circles

Most of us are familiar with Count Buffon's Needle Problem -- the one in which a board of large size is ruled with parallel lines spaced equidistantly, and a needle is dropped randomly. This question from geometric probability involves the constant pi in the answer, the question being to find the probability that the needle touches a line. In this article, the author replaces the parallel lines with concentric circles. One interesting conclusion is that the answer for an infinite number of concentric circles is the same as for the original question. For the case of N circles, however, the computations are quite involved. Naturally, one can create a simulation to approximate pi using the theoretical answer as the expected frequency of line crossings and pi as a unknown. Given the recently reported calculation of pi to over 134,000,000 places, however, such a simulation is more entertaining and pedagogical than valuable as an approximator. But the author's interesting variation is worthwhile in its own right, and what's more, he gives a half-dozen references to other works that either shed light on or vary Buffon's original problem from 1777.