Letting X_i be the half of the scalar product of W with the i-th 
vector V_i, we have n+1 numbers X_0,X_1,...,X_n with the propery 
that X_n = -X_0  and for i=1,...,n,  X_{i+1} equals to X_i+1 or 
to X_i-1. That is, we have a particle on a line at point X_0.
At each step the particle can move one step to the left (-1) or
to the right (+1). Since at the end the particle must reach the
point -X_0, at some step it must  hit the zero point 0.