The object of the following chapters is to give clear and unmistakable instruction on the lines and markings of the hands, both from the student's standpoint and from that of the general reader. This is not usually the course adopted in books p... Read more of The Line Of Head Or The Indications Of The Mentality at Palm Readings.orgInformational Site Network Informational
Privacy
Home Top Rated Puzzles Most Viewed Puzzles All Puzzle Questions Random Puzzle Question Search


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.


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.










Random Questions

Odd And Even Digits.
Money Puzzles
The Hydroplane Question.
Money Puzzles
Bridging The Ditch
THE SQUIRE'S CHRISTMAS PUZZLE PARTY
The Reve's Puzzle
CANTERBURY PUZZLES
Those Fifteen Sheep.
Combination and Group Problems
The Coloured Counters.
Chessboard Problems
An Episcopal Visitation.
The Guarded Chessboard
The Clerk Of Oxenford's Puzzle
CANTERBURY PUZZLES
Sir Edwyn De Tudor.
Money Puzzles
The Suffragists' Meeting.
Money Puzzles
Youthful Precocity.
Money Puzzles
The Coinage Puzzle
THE PROFESSOR'S PUZZLES
Curious Numbers.
Money Puzzles
The Honest Dairyman.
Measuring, Weight, and Packing Puzzles.
The Stop-watch.
Money Puzzles