There is an 8 by 8 chessboard in which two diagonally opposite corners have been cut off.You are given 31 dominos, and a single domino can cover exactly two squares. Can you use the 31 dominos to cover the entire board?

Answer: No

This can be prooved using proof by contradiction that this is not possible. Suppose it were possible to completely cover the modified chessboard with non-overlapping dominoes. Now in any covering, every domino must cover exactly one white square and one black square. Thus the modified board must have exactly the same number of black and white squares. On the other hand, notice that the two removed squares must have been the same color because they came from diagonally opposite corners. Thus there cannot be the same number of white squares and black squares in the modified chessboard!

Therefore it must be impossible to cover the modified board with non-overlapping dominoes!