Particles in a box

This week’s Fiddler is an optimization problem about fitting particles in a box.

You have three particles inside a unit square that all repel one another. The energy between each pair of particles is $1/r$, where $r$ is the distance between them. To be clear, the particles can be anywhere inside the square or on its perimeter. The total energy of the system is the sum of the pairwise energies among the particles. What is the minimum energy for $9$ particles, and what arrangement of the particles produces it?

My solution:
[Show Solution]

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