The issue here is that the exterior doors are included in the count of how many doors a room has. Meaning, for example, if room 1 has some exterior doors, then they will be counted in n_1. Since i need to *prove* that the number of exterior doors in even, i can't assume that is even, and we're only told that every room has an even number of doors, so each n_i for i>0 is even. That means we have no way of knowing n_0 + n_1 + n_2 +...+n_r at this point that is even.
+198 