In this other puzzle, ThomasL asks for three similar pieces which can be arranged to exactly cover all of an 8x8 chessboard, except for a single square — for any of the 64 possible single squares.
I follow this up by asking: Suppose we don't require the pieces to be similar in shape. Can you find another set of three pieces that can be rearranged, rotated, and flipped to exactly cover any 63 of the squares on an 8x8 chessboard?
How many such sets are there? Prove it.
Does the answer change if we permit the pieces to be discontiguousdiscontinuous? (That is, if you're allowed to draw the shapes on transparencies and overlay them, does the problem admit more solutions?)
EDIT: I've just found Universal dissection which seems to be this same problem with a very slightly different question as the puzzle.