# Are You Related to Confucius?

Are all of us descendants of Confucius?  Here’s a curious mathematical argument that suggests just that.

No matter who you are, you came from a mother and a father (I won’t go into details).  So, in your family tree, the part behind you has two branches, like this:

The same goes for your mother and father, and their mothers and fathers, and so on.  Thus, continuing on back the line, you see a family tree like this

And it just keeps going and going and going.  An interesting mathematical feature of this tree is that, as your move backward in time, each generation has twice as many branches as the previous generation, roughly speaking.  Thus, when you go back a hundred or so generations, to the time of Confucius, the number of branches in your family tree is roughly $2^{99}$, or 633,825,300,114,114,700,748,351,602,688 (thanks, WolframAlpha).

A reasonable estimate is that at the time of Confucius there were around 250 million total people in existence.  Each of those $2^{99}$ spots in your family tree has to be filled by someone, which means that, on average, each person in existence at that time had to fill roughly

$\frac {2^{99}} {250,000,000}$ =  2,535,301,200,456,458,802,993

of the spots in your family tree.   It seems like a statistical impossibility that Confucius wasn’t one of them.  So, I guess that makes us cousins?

Categories: ChallengeProbability

#### patrick honner

Math teacher in Brooklyn, New York

#### Sean · October 5, 2010 at 9:00 pm

That is very narrow-minded thinking. This argument deliberately forgets several key factors. The first fact missing is that it is not at all entirely impossible for us to share more than one common ancestor. The second is the fact that many of us are the product of the same race repeatedly in-breeding. If this was not true, then there would be only one race and no different physical characteristics. It is possible that we have an ancestor that goes far beyond the years of Confucius, presumably from Africa.

#### Ivan · October 5, 2010 at 10:23 pm

Wouldn’t the fact that Confucius had 2 kids, and given that even one of them had his/her own kids mean that the population in world right now should be around 3.1e29?

#### Alexander Bogomolny · September 29, 2013 at 8:28 am

There is certainly a logical flaw in the argument. Everything else aside, “statistical impossibility” is not the same “mathematical certainty” which is implied by the caption. More importantly, the same argument shows that each of us is a descendant of all estimated 250,000,000 Confucius’ contemporaries.

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