This problem was asked by Google.

The area of a circle is defined as pi*r². Estimate pi to 3 decimal places using a Monte Carlo method.

Hint: The basic equation of a circle is x²+ y²= r²

I'm going to use Go, a perfectly fine general purpose programming language.

The standard library function rand.Float64() gives back a pseudo-random number in the range [0.0,1.0).

A quarter of a 1.0-unit radius circle fits inside a quarter of a 2x2 square.

By calling rand.Float64() twice, once for an X value, once for a Y value, the program gets a point in the 1.0 x 1.0 quarter of a 2x2 square. The square of the distance from the origin is X*X+Y*Y. If that square of the distance is less than 1.0, the point is inside the quarter of a circle, If the square of the distance is greater than 1.0, the point is outside the quarter of a circle.

If we assume that any such pseudo-randomly chosen point is uniformly likely to be anywhere in the 1.0 x 1.0 quarter of a square, and we choose enough random points, pi will be 4.0 * (inside/total).

I wrote a program to choose a large number of points pseudo-randomly, calculate the square of the distance, and count all the points that lie inside the quarter of a unit circle.

You just need a loop counter, a count of points inside the circle, seed the random number generator, and make a simple floating point calculation.

For 100,000,000 pseudo-randomly-chosen points:

$ time ./pi1
pi = 3.141572
./pi1  4.13s user 0.01s system 99% cpu 4.147 total

A method from 1733, Buffon's needle, can also be used. I wrote a program that computes pi using this method.

$ go build buffon.go
$ time ./buffon 6 5.01 100000000
cracks 6.000000 apart
needle 5.010000 long
100000000 trials
3.141306
./buffon 6 5.01 100000000  6.91s user 0.01s system 99% cpu 6.926 total

Buffon's needle converges on pi far slower than the quarter-circle method above. It's also less satisfying as a method of estimating pi: it includes (at least) using a trig function on each iteration, and trig functions are inextricably linked with pi. My version also includes using a constant 2*pi to when generating the angle that Buffon's needle forms with the horizontal cracks.

This isn't a super hard question, if you've seen Physics Girl try this in real life with darts.

I'm not sure this is a good interview question to find out if a candidate can write a program. The actual program I wrote ended up as 27 lines of leisurely-formatted Go. It has a loop and an if-statement. It makes 2 calls to a library function, and a simple floating point calculation.

It may be a good question if the candidate hasn't seen Physics Girl, or has not seen any of the large number of other demonstrations of this that exist in the world. It requires a grasp of pseudo-random numbers and plane geometry, and maybe a small flash of insight to marry the two.

It's possible that interviewers want candidates to talk about how this method converges on pi, and candidates should discuss that.

Or maybe, since Google asked this, they just want to see if the candidate knows about Buffon's needle.

Another version in python.