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I don't understand why it isn't just the square root of two. Can't the points be arbitrarily close to the corners?
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In that case, where would you put the 5th point?
Let me rephrase the problem, just in case:
The closest two points of five in a square cannot be further than X apart.
Find X.
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That clarification helps. I'm still working on my coffee this morning.
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Visual representation of the problem.
The 5 points were chosen at random here.