Signs and Wonders in Experimental Mathematics :)

Buffon’s Needle experiment, proposed by 18th-century mathematician Georges-Louis Leclerc, also known as the Count of Buffon, is a renowned thought experiment that reveals an unexpected link between randomness and the mathematical constant .

The setup is simple: imagine a floor with equally spaced parallel lines and a needle whose length is equal to or less than the distance between these lines. When you drop the needle, it may land crossing a line or completely between the lines. By dropping the needle repeatedly and counting how often it crosses a line, you can approximate the value of π. The percentage of a crossing event is linked to through a specific formula, and, remarkably, the more times you repeat the experiment, the closer your calculated value will approach .

Fig. 1

Suppose a needle of length is dropped at random onto a floor with parallel lines spaced a distance apart. Let represent the angle between the needle and the lines on the floor, and let be the distance between the midpoint of the needle and the nearest line. For the needle to cross a line when , the following condition must hold:

Due to symmetry, the constrains are and .

Fig. 2

Tossing a needle is equivalent to selecting a point within the rectangle shown in Fig. 2. When the needle crosses a line, by (*), the point falls under the curve .

Since the number of randomly chosen points within a region is approximately proportional to its area, select at random points within the rectangle and let represents the number of points that fall under the curve, we have

That is,

Thus,

which leads to an approximation for as:

An actual experiment was carried out by a diligent Italian mathematician, Mario Lazzarini, who made needle tosses and recorded that of them intersected a boundary line. This experiment, documented in Periodico di Matematica per l’insegnamento secondario, Vol. 17 (1901), produced an approximation for :

This value is accurate to six decimal places!

Excerpts from Lazzarini’s “Un’applicazione del calcolo della probabilita”

Inspired by Mario’s groundbreaking result, I recently embarked on conducting my own version of the experiment. To my delight, the outcomes were nothing short of a wonder:

Total	Intersected
198	105	3.142857142857143
2997	1590	3.141509433962264
3195	1695	3.141592920353983
311979	165510	3.141592653011902

Exercise-1 Explain:

Exercise-2 Explain: