  Quantum games embody non-intuitive consequences of quantum phenomena, such as
entanglement and contextuality. The Mermin-Peres game is a simple example,
demonstrating how two players can utilise shared quantum information to win a
no - communication game with certainty, where classical players cannot. In this
paper we look at the geometric structure behind such classical strategies, and
borrow ideas from the geometry of symplectic polar spaces to maximise this
quantum advantage. We introduce a new game called the Eloily game with a
quantum-classical success gap of $0.2\overline{6}$, larger than that of the
Mermin-Peres and doily games. We simulate this game in the IBM Quantum
Experience and obtain a success rate of $1$, beating the classical bound of
$0.7\overline{3}$ demonstrating the efficiency of the quantum strategy.
