Lab 07: More sophisticated loops 

Goals:
- Understand array syntax
- Manipulate array objects
- Design solutions that involve more complex loops
- Practice testing and documenting more sophisticated programs.

0. Exercises 4.41 - 4.59.

1. Enhance the LogAnalyzer class with a method calles busiestHours that returns
a collection with the busiest hours of a day. This method is similar to the
busiestHour method that you developed before, but busiestHours will return more
than one hour in case there is a tie for busiest hour.

2. Exercises 5.1-5.20.

3. Extra credit
Develop an application that can provide a client with an approximation of pi.
The pi approximator application will approximate pi as follows:

Consider a circle with radius r = 1, inscribed within a (the smallest possible) 
square.  
Notice that the area of the square, as = (2 * r) * (2 * r)= 2 * 2 = 4.
Also notice that the area of the circle, ac=pi * r * r = pi * 1 * 1 = pi,
precisely the value that we are after.

Now consider the following idea:
If we draw the square with the circle on it and "throw random darts" at them,
ratio ac/as is the ratio of darts that fall in the circle.
This ratio is of interest because
pi = ac = ac / as * as = (ac / as ) * as =
     ratio_of_darts_that_fall_in_circle * 4.

So this is our algorithm for computing pi:
Compute the ratio_of_darts_that_fall_in_circle and then multiply by 4.

The ratio_of_darts_that_fall_in_circle can be computed as follows:
Think of the circle on the cartesian plane and centered at position (0,0).
Generate a large number of random coordinates (x,y) 
with the x and y positions in [-1,1] // "dart" positions
and keep track of
the number of "darts" that fall in the circle //(x*x + y*y) <= 1 
and the total number of "darts."
You can compute the 
ratio_of_darts_that_fall_in_circle and then multiply by 4
at any time to approximate pi.
The approximation improves as you "throw" more "darts."

Pedagogical note: You have just done your first Monte Carlo simulation!