THE EIGHT ROOKS.
(
Chessboard Problems)
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FIG. 1.]
[Illustration:
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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.
Answer:
Obviously there must be a rook in every row and every column. Starting
with the top row, it is clear that we may put our first rook on any one
of eight different squares. Wherever it is placed, we have the option of
seven squares for the second rook in the second row. Then we have six
squares from which to select the third row, five in the fourth, and so
on. Therefore the number of our different ways must be 8 x 7 x 6 x 5 x 4
x 3 x 2 x 1 = 40,320 (that is 8!), which is the correct answer.
How many ways there are if mere reversals and reflections are not
counted as different has not yet been determined; it is a difficult
problem. But this point, on a smaller square, is considered in the next
puzzle.
