Fall 2016, problem 20

Suppose that the points $x_1,\dots,x_n$ are all distinct and lie on a line, with distance between any two pair of points given by $d_{ij}=|x_i-x_j|$. Suppose further that the number of times that each distance will occur is at most two, i.e. that any positive real number $r$ is equal to $d_{ij}$ for at most two $1\le i\lt j\le n$. Then what is the least number of distances that occur exactly once, i.e. what is the least number of positive real numbers $r$ such that $r=d_{ij}$ for one and only one choice of $1\le i\lt j\le n$?

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