THE EIGHT QUEENS.
(
Chessboard Problems)
[Illustration:
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]
The queen is by far the strongest piece on the chessboard. If you place
her on one of the four squares in the centre of the board, she attacks
no fewer than twenty-seven other squares; and if you try to hide her in
a corner, she still attacks twenty-one squares. Eight queens may be
placed on the board so that no queen attacks another, and it is an old
puzzle (first proposed by Nauck in 1850, and it has quite a little
literature of its own) to discover in just how many different ways this
may be done. I show one way in the diagram, and there are in all twelve
of these fundamentally different ways. These twelve produce ninety-two
ways if we regard reversals and reflections as different. The diagram is
in a way a symmetrical arrangement. If you turn the page upside down, it
will reproduce itself exactly; but if you look at it with one of the
other sides at the bottom, you get another way that is not identical.
Then if you reflect these two ways in a mirror you get two more ways.
Now, all the other eleven solutions are non-symmetrical, and therefore
each of them may be presented in eight ways by these reversals and
reflections. It will thus be seen why the twelve fundamentally different
solutions produce only ninety-two arrangements, as I have said, and not
ninety-six, as would happen if all twelve were non-symmetrical. It is
well to have a clear understanding on the matter of reversals and
reflections when dealing with puzzles on the chessboard.
Can the reader place the eight queens on the board so that no queen
shall attack another and so that no three queens shall be in a straight
line in any oblique direction? Another glance at the diagram will show
that this arrangement will not answer the conditions, for in the two
directions indicated by the dotted lines there are three queens in a
straight line. There is only one of the twelve fundamental ways that
will solve the puzzle. Can you find it?
Answer:
The solution to this puzzle is shown in the diagram. It will be found
that no queen attacks another, and also that no three queens are in a
straight line in any oblique direction. This is the only arrangement out
of the twelve fundamentally different ways of placing eight queens
without attack that fulfils the last condition.
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