We study the quantum mechanics of a charged particle confined to the surface of a cube enclosing a magnetic monopole. The magnetic field is chosen to have a constant magnitude on each face and to point along the outward normal, preserving the rotational symmetry of the cube. We formulate the continuum problem using two gauge patches on the cube surface and show that consistency of the wavefunction gives the Dirac quantization condition. Since an explicit vector potential does not remain invariant under ordinary rotations, we construct gauge-modified rotation operators and use them to classify the eigenstates. Even monopole charges are described by the irreducible representations of the cubic rotation group $O$, while odd monopole charges require the spinorial representations of the binary octahedral group $2O$. We compute the spectrum with a gauge covariant finite-difference discretization and find Landau-level-like manifolds whose degeneracies are split by the discrete cubic symmetry. We also study the corresponding tight-binding Hofstadter problem on the discretized cube. The resulting spectrum contains the usual magnetic subband structure together with additional gap states localized near the cube corners.