Extensions of Buffon’s Needle Problem Using the n-Stick Star by Isabel Mikheev '25

Extensions of Buffon’s Needle Problem Using the n-Stick Star by Isabel Mikheev ’25, Monday April 14, 1:00 – 1:50pm, North Science Building 113, Wachenheim, Mathematics Thesis Defense

In 1777, Georges-Louis Leclerc, Comte de Buffon introduced one of the earliest problems in geometric probability. Consider randomly tossing a needle of unit length onto a plane ruled by the lines y = n (n = 0, ±1, ±2, …). The probability that the needle intersects some line, given no preference for position or direction, is 2/π. If we consider two of these needles and join them perpendicularly at their midpoints (we call this a “cross”), then the expected number of intersections of the cross and the plane of parallel lines is 4/π. In this talk, we will explore more complex configurations such as the three-stick star and four-stick star, and then generalize our findings to the n-stick star, a group of n needles joined at their centers and spaced at equal angular intervals. We will derive a closed-form expression for the expected number of intersections of the n-stick star and the variance of this estimator for 2/π, and prove that this variance decreases and converges as n approaches infinity. Finally, we will extend this problem to the continuous limit of this construction as the n-stick star approaches a disc, and compute the expected proportion of the disc that will cross the line.