(a) Consider an n × n chessboard, and a rook that is allowed to make the standard moves along the rows and columns. Show that the rook can start at a given square and return to that square after making each of the possible legal moves exactly once and in one direction only [of the two moves (a, b) and (b, a) only one should be made].
(b) Consider an n × n chessboard with n even, and a bishop that is allowed to make two types of moves: legal moves (which are the standard moves along the diagonals of its color), and illegal moves (which go from any square of its color to any other square of its color). Show that the bishop can start at a given square and return to that square after making each of the possible legal moves exactly once and in one direction only, plus n2/4 illegal moves. For every square of its color, there should be exactly one illegal move that either starts or ends at that square.