Bald land
These are the conditions in Baldyville:
-
No two inhabitants have the same number of hairs on their head.
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No inhabitant has exactly 518 hairs.
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There are more inhabitants in town than hairs on any individual inhabitant's head.
What is the highest possible number of inhabitants?
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If there is only 1 inhabitant, then from condition 3 we know he's bald, i.e., he has 0 hairs on his head.
If there are 2 inhabitants, then from condition 3 we know that each of them has either 0 or 1 hair(s) on his head. But by condition 1, they can't be the same, so one has to have 0 and the other has 1.
Then from conditions 1 and 3 we can determine that if there are n villagers, then there must be people with 0 , 1 , 2 , 3 , . . . , n − 3 , n − 2 , and n − 1 hairs on their heads. Then condition 2 cannot hold if there are 519 or more people in town, so there can be no more than 518 inhabitants of Baldyville.