  The Mermin-Peres magic square game is a cooperative two-player nonlocal game
in which shared quantum entanglement allows the players to win with certainty,
while players limited to classical operations cannot do so, a phenomenon dubbed
"quantum pseudo-telepathy". The game has a referee separately ask each player
to color a subset of a 3x3 grid. The referee checks that their colorings
satisfy certain parity constraints that can't all be simultaneously realized.
  We define a generalization of these games to be played on an arbitrary
arrangement of intersecting sets of elements. We characterize exactly which of
these games exhibit quantum pseudo-telepathy, and give quantum winning
strategies for those that do. In doing so, we show that it suffices for the
players to share three Bell pairs of entanglement even for games on arbitrarily
large arrangements. Moreover, it suffices for Alice and Bob to use measurements
from the three-qubit Pauli group. The proof technique uses a novel connection
of Mermin-style games to graph planarity.
