Six degrees of separation - It doesn't work!!!

Well I was just thinking about it yesterday night and I tried to work it out from simple school level mathematics...

For 6 Billion people to know each other through 6 degress means a geometric series

1 + x + x^2 + x^3 + x^4 + x^5 + x^6 = (x^7 - 1) / (x - 1) = 6 Billion People (Total Population)

Here x is the number of unique people each one in the chain knows.

Well using trial and error I find it is between 42 & 43. So let's assume it is 43 People.

You should be on 1st name basis with these 43 people and each of these 43 guyes should be on a first name basis with other 43 people (people not in your list).

Now practically all the people I know have lot of common friends so they won't count. Lets take a simple example

The situation below is when there are no commong friends

----> D
---> B ---> E
A
---> C ---> F
----> G

In this A knows 6 people (B, C, D, E, F, G)) contrast this when there are common friends

---> B ---> D
A
---> C ---> E

In this A knows only 4 people (B, C, D & E).

So we see that common friends reduce the number of people you actually know.

Now count the number of people you know on 1st name basis chances are they are less than 43... Even if you do know then the chances of those guys knowing 43 other people well that's gonna be more difficult.

Therefore 6 degrees of separation doesn't work :)

To put it practically tell me the degrees of separation it will take for you to link with a person in lets say Somalia...