[math-fun] How many pieces? puzzle

Let the n-dimensional torus T^n be defined as the n-cube Q_n = [-1, 1]^n with its opposite faces identified. (That is, any point x of Q_n with coordinate x_k = ±1 for some k is identified with the point having the same coordinates except with x_k changed to -x_k.) The vertices of Q_n are the 2^n points {-1, 1}^n. Now we are going to cut T^n along each of the 2^(n-1) hyperplanes defined as the perpendicular bisectors of each pair of antipodal vertices of Q_n. This requires extending each hyperplane throughout the n-torus. Puzzle: How many pieces do these planes cut the torus T^n into? E.g., for n = 2 there are two pieces. —Dan