For two subspace, one can express the dimension of the sum as $$ \dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim (U_1 \cap U_2).$$
However, the obvious extension to three subspacess fails, in the sense that $$\dim (U_1 + U_2 + U_3) \neq \dim U_1 + \dim U_2 + \dim U_3 - \dim (U_1 \cap U_2) - \dim (U_1 \cap U_3) \\ - \dim (U_2 \cap U_3) + \dim (U_1 \cap U_2 \cap U_3).$$
I showed the above with a counter example but can't seem to understand it intuitively. Why does it fail? In general, is the "obvious extension" true for $\dim(U_1+....+U_n)$ when $n$ is even and false when $n$ is odd? Is it just that two a special case?