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Hairy Inhabitants


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Question

In the city of Funkytown, the following facts are true:
No two inhabitants have exactly the same number of hairs.
No inhabitant has exactly 483,207 hairs.
There are more inhabitants than there are hairs on the head of any one inhabitant.
What is the largest possible number of inhabitants of Funkytown?
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  • Solution

483,207


Looking at a very small number of inhabitants, say 2, the number of hairs on their heads would be zero and one. Extrapolating that to increasing numbers, you find that for each number (n), the distribution of hairs is 0 to n-1. Once you get past 483,207 (to 483,208, for example) and none of the inhabitants have 483,207 hairs, and they all have different numbers of hairs, one of them must have more than 483,207 hairs which violates the fact that there are more inhabitants than hairs on any individual.
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