It's not

*that* curious, throw in a little

calculus and

everything makes sense.
The root of the problem is that we need to find the

probability that a

needle will cross a line. For simplicity, we will assign the

length of the needle and the distance between the lines a value of 1. As the needle is dropped, it can land at any

angle. Assuming the lines run vertically, the maximum

horizontal span of the needle, 1, occurs when the angle between the needle and the vertical is π/2

radians. The minimum span, essentially 0, occurs when the needle

falls parallel to the

lines, so that the angle between it and the vertical is 0.

(If you're not into

geometry, π/2

radians is the same as 90

degrees.)

Applying some

trigonometry, we can easily show that the horizontal span of the needle is given by sin(x) where x is the angle the needle makes with the vertical. The next step is to find the average span, that is, the

average value of the function

sin(x) as x varies between 0 and π/2

degrees. From

calculus, this is defined as:

π/2 /
∫ sinx dx /
0 / π/2 - 0
/

(Or if you don't like

calculus, it is the area under the curve

sin(x) divided by the length of the base of the

curve, in this case π/2.)

Evaluating this

integral, we get

|π/2
-cos(x)| = cos(0) - cos(π/2) = 1
|0

Dividing by π/2 - 0, we get 2/π. This is the average

horizontal span of a

needle dropped on the

vertical lines. Without going into a proof, we can show by common sense that the probability that a needle will touch a line equals its

average span divided by the distance between lines (which we defined as 1, so the

probability is just 2/π). (Just think, as span approaches the distance between lines, this ratio approaches one, or maximum

probability, so as span approaches the

average span, the

ratio goes to average probability.)

By definition, probability is the number of needles that cross a line (c) divided by the total number of needles (t) Therefore:

P = c/t = 2/π

Which can be rearranged to π = 2t/c

As with all probability problems, the more trials involved, the better the estimate becomes.

Note, I didnt mean to ruin the mystery of it...in my

opinion the

beautiful simplicity of this makes the

world an even

*more* interesting place.