A Pi in Your Eye

Our fascination with (pi) makes this puzzle interesting and in this case based on a famous experiment conducted in 1777. 3.14159265358979323846264338327950288419716939937510582097494459+

Moline Missile Proving Grounds conducts drop tests for structural members of a missile system. Stats and Statica Major ( brother and sister) students at Moline Tech work summers at Moline Missile Proving Grounds and note that the drop specimens sometimes cross the floor's parallel grid pattern (see Figure 1). They begin to wonder whether there is a relationship of their crossings to the total drops. They vaguely remember an experiment conducted in the 18th century called Buffon's Needle (affectionately referred to as buffoons noodle by students). They view the pattern of the drops thus far (shown in Figure 1) and count 150 crossings out of 250 drops (only 10% shown for visual discrimination) or a ratio of 0.6 (which would yield a value for of 3.333). Stats and Statica begin to resurrect the theory of this phenomenon. First they note that the test specimens are 2 meters long and the spacing between the vertical patterns is 2 meters.

Figure 1.

Figure 1. Moline Missile Proving Grounds Test Specimens

They then observe three cases of events (Figure 2): (specimens fall randomly in angular orientation and distance of the midpoint from the gridlines)

Figure 2.

Figure 2. Three Cases Observed

Case 1 - if midpoint of specimen is close to a vertical grid and the vertical angle that the specimen makes with the grid is large a crossing (includes contact) occurs. Horizontal projections of the half length specimen are equal or greater than x.

Case 2 - if midpoint of specimen is close to a grid but vertical angle is small no crossing occurs.

Case 3 - if midpoint of specimen is further away from grid the specimen remains within the boundaries of the grids They conclude that if the horizontal component of the half length of the specimen is equal to or greater than x a crossing will occur.

This characterization can be illustrated graphically as shown in Figure 3. Since the fall of the specimen can occur in any direction (360°) they construct a graph with the plot of the midpoint projection versus the angle that it makes with the vertical from 0 to 90° or /2 (we only need to plot 1/4 of the possible events as it is sinusoidal and thus repeats in each quadrant). The projection of the half length on the horizontal is plotted on the ordinate axis and the angle is plotted on the abscissa axis.

Figure 3.

Figure 3. Graph of crossings/no crossings

When the angle of the specimen is 90° (/2) the mid length projection reaches a maximum of 1 M. When the angle of the specimen is 0 the mid length projection is 0. The area of the graph is the length times the width or /2. The area under the sinusoid which represents all the events of crossing is 1( found from integral calculus as explained in the answer). The theoretical likelihood of crossings to total drops is the area of the sinusoid divided by the area of the total rectangle or 1//2 which equals 2/ or 0.6366. In their limited sample, Stats and Statica Major counted 150 crossings in a total of 250 drops or a probability of 0.6. In very large samples closer values to the theoretical limit would be observed. Now Stats and Statica Major formulate the question of: "What is the total mean number of crossings if horizontal grids were superimposed forming square grids?" The horizontal grids would be spaced 2 meters apart.

a. 2(2) /

b. 4/

c. ( 2/)

d. [ (2)] 4/

The answer is b.

The answer is the sum of the mean number of crossings of the vertical grids and the mean number of crossings on the horizontal grids. The mean number of crossings of the horizontal grids is equal to the mean number of crossings of the vertical grids (the view 90 degrees displaced does not alter the mean number of crossings). Thus the total number of mean crossings =2/ +2/ =4/.  Note that this is no longer a probability event as probability cannot exceed 1. This is a computation of the sum two sets of mean crossings.

A simple explanation using integral calculus illustrates the classical Buffon's Needle problem:

The variables x and are uniformly distributed where x is the distance from the midpoint of the needle to the nearest gridline (whose distance can vary from b/2 to 0). can be any value from 0 to 360° or 2. W e need only examine the events in the first quadrant as they repeat in the other quadrants. We are seeking the mean value of x/b/2 where x=(a/2)sin. The mean value of (a/b) sin is determined by performing an integration of the following expression: (here we will find that the integral of sinq from 0 to /2 is 1 which we avoided in the statement of the puzzle for a non calculus approach when we assumed that the area of the sinusoid was 1 between the given limits).

When the length of the needle (a) equals the width (b) of the spacing of grids the solution becomes 2/. The denominator /2 assures that the probability is one when is between 0 and /2. When the length of the needle or specimen equals the spacing of the grid width we can easily obtain the value of as:

Programs based on Monte Carlo have been written in DOS and Java to simulate this experiment and have achieved remarkably close attainments of the value of .

Georges-Louis Leclerc, Compte de Buffon lived between 1707-1788. He was born in a family of means and received his excellent education at a Jesuit College, the Universities of Dijon and Angers. He was trained as a lawyer but preferred medicine, botany and mathematics. Buffon's brilliant insight in mathematics enabled him to arrive to this remarkable observation of the relationship of to the tossing of test specimens whether needles crossing parallel grid lines or French stick loaves as they crossed the lines between tiles laid on a floor (the latter as legend has us believe) . He was a prolific author who compiled a popular thirty-six volume encyclopedia Histoire Naturelle. He conducted experiments on the establishing the age of the Earth by heating two dozen one inch sized globes to a white glow. The elapsed time from when the test globes glowed white to the moment they could be touched was used to extrapolate the age of the Earth. Buffon concluded that the Earth was nearly 75,000 years old; Newton had earlier concluded that the Earth was 50,000 years old.

References

Beckmann, Petr. A History of . New York: St. Martin's Press, 1971

Boorstin, Daniel J. The Discovers. New York: Vintage Books A Division of Random House, 1985

Dorrie, Heinrich. 100 Great Problems of Elementary Mathematics Their History and Solution. New York: Dover Publications, Inc., 1965

Gamow George. One Two Three…Infinity. New York: Mentor Books, 1960

Mosteller, Frederick. Fifty Challenging Problems in Probability with Solutions. New York: Dover Publications, Inc., 1987

Solkonikoff, I. S. and Redheffer, R. M. Mathematics of Physics and Modern Engineering. New York: McGraw Hill Book Company, Inc. 1958