Consider a system of $2,3,4,$ and $5$ particles. How many pairwise particle$-$particle interactions will there be$?$ What about $10$ or $100$ particles? Consider a system with a density of $1.$ Using this density, how many particles are in a sphere with radius $1,2,5,$ and $10$ units? $($These are math problems, you don$$t have to use programming to solve them, but you can.$)$

Can you think of the benefits associated with using a cut$-$off?

We can modify our calculate$\_$total$\_$energy function to take in a cut$-$off value:

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