Don't I know you?

I got out my Christmas card list the other day, and this started me off on a chain of thought.... Geometric progressions can produce some surprising results. Imagine for example, that you had a certain-sure way of doubling your money every day, and that you started off on the 1st January with 1p, and using your method by the 2nd you had turned it into 2p, by the 3rd into 4p, by the 4th 8p, and so on. How long would it take you to become a millionaire? Months? Years? Decades? Well, the answer is just 28 days! Amazing but true - get your calculator out and convince yourself. So where does my Christmas card list come into this - well I was totting up the total number of people on there (not just the number of cards I send), and it comes to over 100 - and I'm not a particularly social animal, so I would imagine that most adult people could probably list 100 or more people they know. Have you heard of the Six Degrees of Separation? This is a theory that says that you are connected to every other person in the world by at most six other people - that is to say that, if you pick any person in the world at random, you know somebody, who knows somebody, who knows somebody, who knows somebody, who knows somebody, who knows somebody who knows that person. Seems unlikely, but it's all to do with the power of geometric progressions. If we assume that everybody knows 100 people, then you know 100 people who each know 100 people who each know 100 people, and so on.... By the time we get six people away we have connected with 100^6 people, that is 1,000,000,000,000 or 1000 billion (American). Clearly there will be some duplications here - for example some of the 100 people the people you know know will almost certainly also be part of your 100, so we can perhaps reduce this figure to something under 10 billion, but as the population of the world is only 6.5 billion, you can see that the idea of Six Degrees of Separation is quite feasible. And now back to writing my Christmas cards.....