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We take the unit square and have it divided by $n$ lines which are chosen randomly. We choose the lines as follows, choose one of the four sides of the square at random and then choose a random point on the square. Then choose a random point from one of the other three sides and connect the dots to create a line. Do this $n$ times.

What is the expected largest piece of area after $n$ trials?

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    $\begingroup$ Let me try a guess, at least: $c\log n/n^2$. You are putting $2n$ points around the edge of the square. The largest gap will be something like $\log n/n$. Since there are $n$ lines, it is plausible that the height of the shape will be of order $1/n$. $\endgroup$
    – Anthony Quas
    Commented Oct 21, 2017 at 18:42

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