The Buffon Needle Drop: a Math Activity for Pi Day

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Portrait of Georges-Louis Leclerc, comte de Buffon : Image by Musée Buffon à Montbard

Named for Georges-Louis Leclerc, comte de Buffon, the Buffon’s Needle Drop problem estimates the value of pi with a simple experiment. Celebrate Pi Day on March 14 (3.14) by testing Buffon’s Needle.

Why Experiment with Buffon’s Needle?

The Buffon’s Needle experiment combines a fundamental mathematical constant, pi, with the math of calculating probabilities and the process of scientific experimentation.

What is Pi Day? Why Celebrate Pi Day?

Since pi’s value is approximately 3.14159…, Pi Day is celebrated on 3.14, or March the fourteenth.

Pi is the irrational number that relates the circumference of a circle to its diameter, per the equation “pi = [pi] = C/d = 2*r/d”. (‘C’ is the circumference, ‘d’ the diameter, and ‘r’ the radius of a circle).

Pi is an important part of many branches of mathematics, from geometry and trigonometry through calculus and beyond.

The Theory behind Buffon’s Needle

This experiment began with gamblers betting whether an object dropped onto a tiled floor would land entirely within one tile or cross a line between tiles. Buffon realized that the ratio of the size of the object versus the tile would provide the probability of “inside” or “across”.

Start by considering a carpet with parallel lines or strips. Suppose you drop needles of different lengths onto the carpet, but ensure they always are perpendicular to the carpet’s lines.

If the needle is longer than the distance between the parallel lines, then it will always cross at least one line. Therefore the probability of “across” is exactly one.

If the needle were infinitesimally short, it is almost certain to land between the lines, so the probability of “across” is nearly zero.

In general, the probability that the needle crosses a line is the length of the needle divided by the distance between the lines, but limited to the range from zero to one.

Our experiment is different in two ways. First, all the needles or pencils are the same length. Second, we won’t drop them carefully to land perpendicular to the parallel lines; they can land at any angle.

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