Question

Two diagonally opposite corners of an 8 by 8 checkerboard are removed. Can the remaining figure be divided into 31 rectangles, each consisting of exactly two of the original 64 squares that share a common edge?

Answer 1

http://en.wikipedia.org/wiki/Mutilated_chessboard_problem

Answer 2

It cannot be done. The diagonally opposite square are either both white or both black. Removing them leads to 30 black and 32 white squares OR 30 white and 32 black squares which cannot be joined as a rectangle with a common edge.