Given that there are more people living in Dublin than there are 
hairs on the head of any Dubliner, and that no Dubliner is totally 
bald, does it necessarily follow that there must be at least two 
Dubliners with exactly the same number of hairs? 

Here is a variant of the same problem: on the island of Alopecia, 
the following facts are true: 
   1. No two islanders have exactly the same number of hairs. 
   2. No islander has precisely 450 hairs. 
   3. There are more islanders than there are hairs 
      on the head of any one islander. 
What is the largest possible number of islanders on Alopecia?
 
 
 
 

                  THE SOLUTION

The answer to the first question is yes. Assume there are exactly
one million people living in Dublin. If each inhabitant had a
different number of hairs, then there would be one million different
positive whole numbers each less than one million-which is impossible.

The answer to the second problem is 450. Assuming that one islander
is completely bald, there are 450 variants between 0 and 449 hairs;
but, since we know that no islander has 450 hairs, from 451 hairs
onwards the number of variants is the same as the maximum number of
hairs, which means that the third fact stated in the question would
not hold good for any number of inhabitants higher than 450.
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