*Aperiodical guest author Andrew Taylor writes about an intriguing piece of number theory which turns out to also be something else.*

How many ways are there of writing some natural number $n$ as the sum of two squares?

$$ n = p^2 + q^2 $$

I don’t want an answer for some particular $n$. I don’t even want a general formula. I just want to know… *on average*.