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If 5 points are randomly positioned in a unit square, no two points can be greater than square root of 2 divided by 2 apart; divide up the unit square into four squares, and, based on the pigeonhole principle, five points (pigeons) fitting into four squares (holes) means that no two points can be greater than square root of 2 divided by 2 apart.

What about for a random natural number of x points in this unit square?

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    Woops, let me rewrite this...2012-01-23

1 Answers 1

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This is a very difficult problem. For some information, please see Croft, Falconer, and Guy's Unsolved Problems in Geometry.

There you will find a table for the maximum value of the minimum distance for $x$ up to $27$. (Only $11$ of the entries are exact expressions.) The asymptotic formula $2^{1/2}3^{-1/4}x^{-1/2}$ is mentioned, along with further literature references.