I was solving the following problem:
"Find $\angle A + \angle B + \angle C$ in the figure below, assuming the three shapes are squares."
And I found a beautiful one-liner using complex numbers:
$(1+i)(2+i)(3+i)=10i$, so $\angle A + \angle B + \angle C = \frac{\pi}{2}$
Now, I thought, what if I want to generalize? What if, instead of three squares, there were $2018$ squares? What would the sum of the angles be then? Could I make a formula for $k$ squares?
Essentially, the question boiled down to finding a closed form for the argument of the complex number
$$\prod_{n=1}^{k}{(n+i)}=\prod_{n=1}^{k}{\left(\sqrt{n^2+1}\right)e^{i\cot^{-1}{n}}}$$
This we can break into two parts, finding a closed form for $$\prod_{n=1}^{k}{(n^2+1)}$$ and $$\sum_{n=1}^{k}{\cot^{-1}{n}}$$
This I don't know how to solve.