6.3 Dominos: There is an 8x8 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? Prove your answer (by providing an example or showing why it's impossible).
Not possible. There are 8x8 = 64 - 2 spots. 31 dominos accumpy 31 x 2 = 62 places. But each domino must occupy a white a black square. If for a 64 piece board there are 32 of white and black, for a 62 board there are 30 of one and 32 of the other. Because these are not equivalent and the dominos must occupy both of equivalent amounts, this is not possible.