Some problems can be solved by utilizing particular color schemes. Such method does not require advanced math knowledge, therefore can be a good way to train students' problem solving skills. This exercise contains $8$ such problems. Here are three 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. |

13 pages