Buffon's Problem ... Calculating the value of PI using toothpicks

(Also known as 'Count Buffon's Problem')

Imagine dropping a handful of toothpicks onto a sheet of paper that has been ruled with a number of parallel, equidistant lines. What is the probability that any toothpick will land in such a way that it intersects a line?
At first glance, this problem seems difficult to solve. We will attempt to work out that probability, and show how you can find the value of PI by recording how many toothpicks actually intersect a line!

Junior High classes might want to visit our 'Calculating Pi' page, where we outline how you can work out the value of Pi yourself using toothpicks, and send us your data. Or try out our Buffon Applet, which will let you run the experiment online and get some immediate results!
What follows below is an explanation of where the formula comes from, designed for Math 31 students and teachers.

Clearly, the toothpicks can land on the paper in many ways.

However, the position of any toothpick can be characterized by:

the distance of the center of the toothpick from a line
and
the angle formed with the direction of the lines

Clearly, the toothpick will intersect a line if the projection of half the toothpick length on the vertical direction is greater than distance d.

For example:

Projection length exceeds d,
so the toothpick hits a line.

Projection length less than d,
so there is no intersection.

In order to actually calculate the probability that a toothpick will cross a line, let's look at a specific case. We'll make both the distance between the lines, and the length of a toothpick, each exactly 2 inches. (We use non-metric units here for simplicity, and to approximate the actual length of a toothpick.)

Now, we can plot on a grid all the possible toothpick angles (these range from 0 degrees to 90 degrees (

)), against all the possible projection lengths (these range from 0 to 1 inch)

The graph would look like this:

This is a graph of the sine function! Of course, since the 'projection length' is the opposite side of a right triangle for which the hypotenuse is 1, so the projection length is just the sine of the angle:

So the y axis of the graph above is just 'sine' ... it's a sine graph.

We stated that the toothpick would intersect a line if the projection length was greater than the distance from the center of the toothpick to the line. Or, if the toothpick centre is less than the projection length, the toothpick will cross a line.
This can now be reworded as:
"All toothpicks whose centres are at a distance
less than the sine, will cross a line"
These toothpicks are represented by the green region on the graph. All toothpicks that fall in this region, under the sine curve, will intersect a line.
This means that:

The area under this portion of the sine curve (in green):

can be calculated by using the method for finding the area under a curve, called 'integration':

So the area under the sine graph is 1. This means the probability of intersection can be restated as:

This surprising result, where Pi appears as a probability,
can be used to determine the value of Pi!
Let's now generalize these results.

toothpick length ................ c
line separation ................... a
number of toothpicks ......... N
number of intersections ..... M
(c must be less than a)

(We leave it to the Math 31 student to redo the calculations using c and a, to show where this formula comes from. Give it a try ... it's not very difficult.)

This formula was used to predict a value for PI by a turn-of-the-century Italian mathematician named Lazzarini. He made 3408 trials, and found 2169 intersections. Using the formula, he determined Pi correct to six decimal places:

Lazzarini's value: 3.1415929
The actual value: 3.1415926...

There is, however, some speculation that Lazzarini may have 'fudged' his data, since this is a remarkably accurate value for so few trials.
I have done this experiment several times with Junior High math classes, using as many as 1000 trials at once; we were lucky to get 3.1416!

What we'd like to do here is conduct an on-line collection of data from your experiments! Visit our 'Calculating Pi' page, designed for Junior High classes, but open for anyone to contribute to. Conduct the experiment, and send us your results via the form on that page. We'll list all results entered, and keep a running tally on the value of PI generated! We also have a Buffon Applet which will let you try the experiment online and get immediate results.

Calculating Pi | Displayed Results | Buffon Applet