ColoringMethod BrainTeaser Intermediate
$\textbf{Cover the Board}$
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.
$\textbf{Answer}$
No.
$\textbf{Analysis}$
To show this, let's color the board in the following way:
There are $32$ black squares and $30$ white squares. Their counts are different. However, regardless of how to place a small piece, it will always cover one black and one white square. Hence, thirty-one small pieces will cover $31$ black and $31$ white squares. This means that the goal is unachievable.
$\textbf{Note}$
More than often, the answer to such problems is no. This is because if the answer is yes, then the problem will become less interesting. Providing the answer is no, the key to solve this type of problems becomes how to find a neat explanation. One of the most used techniques is the coloring method which is illustrated here.