with (pi) makes
this puzzle interesting and in this case based on a famous experiment
conducted in 1777. 3.14159265358979323846264338327950288419716939937510582097494459+
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.
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)
2. Three Cases Observed
- 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.
- if midpoint of specimen is close to a grid but vertical angle is small
no crossing occurs.
- 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.
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.
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.
The answer is
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/
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:
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:
on Monte Carlo have been written in DOS and Java to simulate this experiment
and have achieved remarkably close attainments of the value of .
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.
A History of . New York:
St. Martin's Press, 1971
J. The Discovers. New York: Vintage Books A Division of Random
100 Great Problems of Elementary Mathematics Their History and Solution.
New York: Dover Publications, Inc., 1965
One Two Three…Infinity. New York: Mentor Books, 1960
Fifty Challenging Problems in Probability with Solutions. New York:
Dover Publications, Inc., 1987
I. S. and Redheffer, R. M. Mathematics of Physics and Modern Engineering.
New York: McGraw Hill Book Company, Inc. 1958