October 28th, 2007, 3:24 pm
Yes. The simplest rationale is that people would have to have at least 6 billion hairs (and be perfectly ordered on the 0 to 6 billion number line) to avoid an overlap. Humans ont have 6 billion hairs, so multiple people must share the same hair count.A variant of the problem is: how small a town would have a high chance that two adults in the town have the same number of hairs? This is a variant of the shared-birthdays-in-a-room problem -- although the chance that a given pair of people in a room share the same birthday is 1/365, the O(n^2) number of possible pairings means the chance of at least one pair sharing a birthday is quite high once the number of people reaches more than a couple dozen. If N is the average number of hairs on the adult human head in that town, I would expect that any population of O(sqrt(N)) adults would have at least one pair with the same number of hairs.