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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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are you asking about the expected greatest distance? –  Mark Jan 23 '12 at 16:35
    
Greatest distance possible. Sorry, that was unclear. –  David Faux Jan 23 '12 at 16:36
    
I don't understand this question. If five points are randomly positioned in a unit square, then the maximum distance between any two of the points is a random variable that can assume any value between $0$ and $\sqrt{2}$. What does the pigeonhole principle have to do with that? (I assume you mean to say something about the largest value that the minimum distance between any two points can have... and I don't get the impression that you're actually asking about anything "random".) –  mjqxxxx Jan 23 '12 at 16:36
    
Woops, let me rewrite this... –  David Faux Jan 23 '12 at 16:40
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up vote 3 down vote accepted

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.

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