Chubby makes nonstandard checkerboards that have $31$ squares on each side.  The checkerboards have a black square in every corner and alternate red and black squares along every row and column.  How many black squares are there on such a checkerboard?
$\textbf{(A)}\ 480 \qquad\textbf{(B)}\ 481 \qquad\textbf{(C)}\ 482 \qquad\textbf{(D)}\ 483 \qquad\textbf{(E)}\ 484$
There are 15 rows with 15 black tiles, and 16 rows with 16 black tiles, so the answer is $15^2+16^2 =225+256= \boxed{\textbf{(B)}\ 481}$
Note: When solving $16^2$+$15^2$, you only need to calculate the units digit.