  We study linear constraint system (LCS) games over the ring of arithmetic
modulo $d$. We give a new proof that certain LCS games (the Mermin--Peres Magic
Square and Magic Pentagram over binary alphabets, together with parallel
repetitions of these) have unique winning strategies, where the uniqueness is
robust to small perturbations. In order to prove our result, we extend the
representation-theoretic framework of Cleve, Liu, and Slofstra (Journal of
Mathematical Physics 58.1 (2017): 012202.) to apply to linear constraint games
over $\mathbb{Z}_d$ for $d\geq 2$. We package our main argument into machinery
which applies to any nonabelian finite group with a ''solution group''
presentation. We equip the $n$-qubit Pauli group for $n\geq 2$ with such a
presentation; our machinery produces the Magic Square and Pentagram games from
the presentation and provides robust self-testing bounds. The question of
whether there exist LCS games self-testing maximally entangled states of local
dimension not a power of $2$ is left open. A previous version of this paper
falsely claimed to show self-testing results for a certain generalization of
the Magic Square and Pentagram mod $d\neq 2$. We show instead that such a
result is impossible.
