The Coloring Method Basic

Examples

 (2744) $\textbf{Cover the Board}$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.

 (2745) $\textbf{Cover the Board (II)}$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?

 (2747) Show that among any $6$ people in the world, there must exist $3$ people who either know each other or do not know each other.