Pigeonhole Theory And Hair

I always find it interesting when someone comes up with a new use for an older theorem.    The pigeonhole theory or dirichlet's principle was applied to the question of "Can two people have  the same number of hairs?" and came up with the answer "yes". This theorem was first proposed back in 1622 and says that when you want to split a certain number of objects among a specific number of drawers, you will end up  with several objects in each drawer.

Although this theorem first appeared in a book published by Jean Leurechon in 1622, it was attributed to Peter Gustav Lejune Dirichlet who lived almost 200 years later.   Although the theorem is rather simple, it is used to explain more complex situations and relationships such as if you have five points arranged randomly on a sphere, four points end up in the same hemisphere. 

A person decided to apply this theorem to the question of "Can two people have the same number of hairs on their heads."  First one has to find out the maximum number of hairs that can be found on a head and the approximate number of people on the earth.  Most people have a total number of hair strands falling between 90,000 and 150,000. The approximate population world wide is around eight million people.  Thus there should be some people who have the exact same number of hairs based on this theorem.

Furthermore, one can assume that if you have a million rooms and  all eight million people have the same amount of hair.  This means that everyone will be in one room and the other 999,999 will be empty.  On the other hand, if people divide themselves up so that the minimum number end up in each room, how many would that be?  I believe that ends up as around 8,000 people per room. 

Another example might be the question of how many people share the same birthday in New York City. We know that some will based on this theory. If you take the population of New York City and divide it by 366 days, you get 8.5 million/366 and end up with 23,000 people who share the same birthday everyday.

Notice the conclusions that people come up using the pigeonhole theory is always based on simple assumptions.  Let me know what you think, I'd love to hear.  Have a great day.