All hyperbolic Coxeter n-cubes

Beside simplices, n-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter n-cubes are not classified. We show that there is no hyperbolic Coxeter n-cube for n≥6, and provide a full classification for n≤5. Our methods, which are essentially of combinatorial and algebraic nature, can be (and have been successfully) implemented in a symbolic computation software such as Mathematica®.