[Illustration:
+---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | R | R | R | R | R | R | R | R | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+
FIG. 1.]
[Illustration:
+---+---+---+---+---+---+---+---+ | R | | | | | | | | +---+---+---+---+---+---+---+---+ | | R | | | | | | | +---+---+---+---+---+---+---+---+ | | | R | | | | | | +---+---+---+---+---+---+---+---+ | | | | R | | | | | +---+---+---+---+---+---+---+---+ | | | | | R | | | | +---+---+---+---+---+---+---+---+ | | | | | | R | | | +---+---+---+---+---+---+---+---+ | | | | | | | R | | +---+---+---+---+---+---+---+---+ | | | | | | | | R | +---+---+---+---+---+---+---+---+
FIG. 2.]
It will be seen in the first diagram that every square on the board is either occupied or attacked by a rook, and that every rook is "guarded" (if they were alternately black and white rooks we should say "attacked") by another rook. Placing the eight rooks on any row or file obviously will have the same effect. In diagram 2 every square is again either occupied or attacked, but in this case every rook is unguarded. Now, in how many different ways can you so place the eight rooks on the board that every square shall be occupied or attacked and no rook ever guarded by another? I do not wish to go into the question of reversals and reflections on this occasion, so that placing the rooks on the other diagonal will count as different, and similarly with other repetitions obtained by turning the board round.
