### The Buffon Needle Problem Revisited in a Pedagogical Perspective

Ivar Johannesen
2009 The Mathematica Journal
Imagine marking the floor with many equally spaced parallel lines and a thin stick whose length exactly equals the distance L = 1 between the lines. If the stick is thrown on the floor, the stick may or may not cross one of the lines. The probability for a hit will involve Π. This is surprising since there are no circles involved, on the contrary all is typically linear. If we repeat the experiment many times, and keep track of the hits, we can get an estimate of the irrational number Π. We
more » ... al number Π. We also consider sticks of length L > 1. This exercise can easily be done in a first year calculus course, where the students are challenged to consider concepts such as probability, definite integral, symmetry and inverse trigonometric function. The solution to this problem will therefore give many applications in a variety of fields in calculus. We go on throwing regular polygons of different sizes, increasing the number of edges and at last reach the ultimate goal: throwing circular objects. This paper illustrates the process of throwing sticks, polygons and circles analytically and graphically, and carrry out calculations for different n − gons. The result always include the number Π, except when the circle is introduced! We will also see the circle result as a limiting value when n increases to infinity. This paper illustrates the process of throwing sticks, polygons and circles analytically and graphically, and carrry out calculations for different n − gons. The mathematics necessary is elementary and suitable for students in a first calculus course. The students will solve the necessary integrals and calculate the probabilities by hand before invoking Mathematica. The introductory part of the lab considers sticks of length L = 1, the same unit length as Avignon, June 2006 8th International Mathematica Symposium