One family of nonlocal games that has received considerable attention is the family of linear constraint system (LCS) games, where Alice and Bob try to convince the referee that a linear system $Ax = b$ has a solution. The perfect classical, quantum and quantum commuting strategies of an LCS game are encoded in a finitely presented group, called the solution group of the linear system. Many results are know about LCS games over $\mathbb{Z} / 2\mathbb{Z}$, ranging from simple pseudotelepathy scenarios such as the Mermin-Peres magic square to advanced results about undecidablility and separations between different correlation sets. By contrast, over $\mathbb{Z} / d\mathbb{Z}$ for $d$ odd, only basic results are known, and it was recently conjectured by Chung, Okay and Sikora that no pseudotelepathy scenarios exist in this setting. In this talk, I will use graph theory to explain why it is harder to get pseudotelepathy modulo odd numbers, and I will present the first known examples of pseudotelepathy in LCS games modulo odd primes. This talk is based on joint work with David Roberson and runs parallel to recent work of Slofstra and Zhang.