On Magic Finite Projective Space

This paper studies a generalization of magic squares to finite projective space $\mathbb{P}^n(q)$. We classify at all functions from $\mathbb{P}^n(q)$ into a finite field where the sum along any $r$-flat is $0$. In doing so we show connections to elementary number theory and the modular representation theory of $\operatorname{GL}(n,q)$.