Math Puzzle - THE EIGHT ROOKS.

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THE EIGHT ROOKS.

(Chessboard Problems)
[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.

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