Buffon’s Needle Problem.
Buffon’s needle Problem, which was first posed in 1777 by its namesake Georges-Louis Leclerc, Comte de Buffon, is a fantastically simple problem which allows us to calculate, or at least approximate, the value of π through a simple experiment. It can be stated in many different formulations, but we will consider the following one:
Suppose a needle of length L is dropped onto a piece of paper which has been lined with evenly spaced parallel lines. What is the probability of that the needle will cross one of these lines?
At first glance it may seem that such a simple puzzle could have nothing to do with approximating a number as strange as π. This scepticism is more than understandable. So let me try and make you a believer. You will need only two things for this problem:
1) A basic understanding of Trigonometry.
2) A basic understanding of how to calculate definite integrals of trigonometric functions. (If you don’t have this, don’t worry, you will still be able to follow almost all of the solution and will just have to trust me on one calculation).
I should say that you will also need a very brief introduction to probability theory, but don’t worry, where it’s relevant I will provide it.
So, without further ado, let’s approach this problem. A very common technique employed by mathematicians, when approaching a problem they are unsure how to solve immediately, is to begin by considering a particular case. So, with this in mind, suppose we consider the case in which we drop our needle and it falls in between two of the lines on the paper, see the below figure, in which, the parallel red lines represent two lines on the page and the black line is our needle.
Let’s attach some quantities to this scenario, all of which are shown in the next figure. We have already said that the length of the needle is L. Let is denote the distance between the evenly spaced parallel lines as D and let us take X to be the perpendicular distance from the lower end of the needle to the next line above this end. Next, let’s draw a line, which is parallel to the lines on the page, through the lower end of the needle. This will form an acute angle between this newly drawn line and needle, let’s call this angle θ. Finally, let’s draw a line from our newly drawn line to the top of the needle, and lets draw this at an angle of 90° to the newly drawn line. This will give us a right angle triangle and basic trigonometry now tells us that the opposite side, of the triangle we just formed, will have length L sin(θ).
This will be how we set up our problem no matter where the needle falls. For example, if the needle was to cross one of the lines, our set up would look like:
and if the needle was to fall so that the left end of the needle was higher than the right, our set up would look like one of the following:
Let us now make note that, from our set up, it is clear that if the needle does not cross any of the lines, then we must have that the distance Lsin(θ) ≤ X, and if the needle does cross any of the lines, then we must have that the distance Lsin(θ) > X.
We are now in a position to conduct our experiment. Notice that L and D are constants each time we drop our needle. It is θ and X that our variables, so these are what need to be recorded from each drop. However, it should be clear that the relationship between L and D has a big effect on the outcome of our experiment. In particular, suppose L=1 m and D=0.01 m (i.e 1 cm). Then it would seem that dropping our 1 metre long needle we are very like to cross one of the lines that are 1 centimetre apart. What about the other way around? In this case it would seem that dropping our 1 centimetre long needle we are very unlikely to cross one of the lines that are 1 metre apart. Because of this relationship having such an evident effect on our experiment we will split our experiment into two cases. Case one is when L ≤ D and case two is when L >D.
Recall, it was the value of the function L sin(θ) that told us whether our needle crossed a line or not. Furthermore, since L is constant, it is actually the value we measure for θ that determines the value of this function. Let us denote this function by F(θ)= L sin(θ). To see what kind of effect the value of θ has, let’s draw a graph of this function as the value of θ varies.
In particular, let’s consider case one, when the needle length is shorter than the distance between the lines, and let us now consider a graph of the potential needle length, L, versus the acute angle formed, θ. On this graph let us sketch the curve of the function F(θ)=L sin(θ), which recall tells us, for each value of θ, whether or not we crossed a line. This graph is shown below, for a randomly placed L< D.
As we already mentioned before, if the needle does not cross any of the lines, then we must have that the distance L sin(θ) ≤ X, and if the needle does cross any of the lines, then we must have that the distance L sin(θ) > X. In terms of our graph this says that if θ is measured so that the point (θ,L) lies below the graph of the function F(θ), and so is in the shaded region, then the needle must have crossed a line on the paper. If θ is measured so that the point (θ,L) lies above the graph of the function F(θ), and so is not in the shaded region, then the needle can not have crossed a line on the paper. Thus the probability of the needle crossing a line is the ratio of the shaded area to the area of the entire rectangle.
Returning to our problem we now realise that the probability of the needle crossing a line is the ratio of the shaded area to the area of the entire rectangle, i.e.
We have almost complete our problem. The last thing for us to do is to calculate the area of shaded region and the area of rectangle. The area of the rectangle is a simple task and is clearly
As for the area of the shaded region, this is similarly an easy enough task if you know your definite integrals of trigonometric functions.
So, since it can be shown that the area of the shaded area is L we now have:
Great, so how does this allow us to estimate the value of π you ask. Well here is your final lesson in probability.
Returning to our problem and we realise that, provided we are willing to allow for a little error, we can calculate the empirical probability of the dropped needle crossing a line on the page and use this in the formula that we worked above, which recall was
In particular, to calculate P(needle crossing a line) we could drop our needle, let’s say n times, count up how many times the needle did cross the line, let’s say this happens k times, and then use our formula for above to say that
Based off the law of large numbers, the more times we drop our needle, i.e. the bigger n gets, the closer our probability will get to the theoretical value. You should try doing this yourself. I took a blank sheet of A3 paper and lined it with parallel lines evenly space at 6cm. I then dropped a 2 inch (which is 5.08 cm) sowing needle two hundred times. I counted that the needle crossed a line 107 out of 200 times. So from my experiment I can see that I have the following quantities:
Note I said P(needle crossing a line) ≈ 107/200, and not P(needle crossing a line) = 107/200, because, as we’ve discussed, this is our empirical probability which is only an estimate for our theoretical probability. So, rearranging our equation, we have find that
So we can see that we have gotten a relative good estimate for π ≈ 3.14159. In particular we have got an estimate that is correct to two decimal places and is quite close in the third decimal place, which really is not too shabby considering it all came from dropping a needle on some lined paper. If we wanted an more accurate estimate we could always just increase the number of times we dropped the needle.
One can do this exact same thing on the case that L>D, but it does take a little more work. Since the purpose of this piece was just to show how to calculate such an estimate of π, we will not show the other method and instead suggest the interested reader search for it instead. One such place to find it is in a well written article by Lee L. Schroeder¹.
(1) Lee L. Schroeder, Buffon’s needle problem: An exciting application of many mathematical concepts, The Mathematics Teacher, Vol. 67, №2, (1974),183–186.