$\textbf{Cover the Board}$

Joe cuts off the top left corner and the bottom right corner of an $8\times 8$ board, and then tries to cover the remaining board using thirty-one $1\times 2$ smaller pieces. Is it possible? Note: a smaller piece can be rotated, but cannot be further broken up.

$\textbf{Cover the Board (II)}$

Joe cuts off a $2\times 2$ corner from an $8\times 8$ board, and then tries to cover the remaining part using $15$ L-shaped grids made of $4$ grids as shown. Is it possible?