You place two stones on a chessboard and start moving them one by one, every time shifting a stone to an adjacent cell - left, right, up, or down, without letting the stones at any time occupy the same cell. Is it possible to find such sequence of moves, that all possible combinations for the position of the stones occurs exactly once?
No, such sequence does not exist. Call a position of the stones "matching" if the stones occupy cells of the same color, and "opposite" if the stones occupy cells of opposite colors. If such sequence exists, then the number of matching and opposite positions must differ by at most 1. However, the number of matching positions is 64x31, whereas the number of opposite positions is 64x32, so this is not true.