As part of an exercise out of Hatcher's Algebraic Topology I want to compute the homology of an n-torus by cellular homology, where we've defined the torus to be the square $X = [-1,1]^n$ with pairs of opposite faces $X \cap {x_i = +-1}$ for $ i = 1, 2, ... ,n$ identified by reflection in the plane ${x_i = 0}$.
I can "see" that one CW structure of the n-torus is binomial, $ \binom{n}{k}$ for the number of k-cells etc. And I've drawn appropriate diagrams for the 1, 2, and 3-tori to show this. But I'm not sure how to prove it for the other naturals. Can anyone offer a hint or some guidance?