BISHOPS IN CONVOCATION.
(
Chessboard Problems)
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]
The greatest number of bishops that can be placed at the same time on
the chessboard, without any bishop attacking another, is fourteen. I
show, in diagram, the simplest way of doing this. In fact, on a square
chequered board of any number of squares the greatest number of bishops
that can be placed without attack is always two less than twice the
number of squares on the side. It is an interesting puzzle to discover
in just how many different ways the fourteen bishops may be so placed
without mutual attack. I shall give an exceedingly simple rule for
determining the number of ways for a square chequered board of any
number of squares.
Answer:
The fourteen bishops may be placed in 256 different ways. But every
bishop must always be placed on one of the sides of the board--that
is, somewhere on a row or file on the extreme edge. The puzzle,
therefore, consists in counting the number of different ways that we
can arrange the fourteen round the edge of the board without attack.
This is not a difficult matter. On a chessboard of n squared squares 2n - 2
bishops (the maximum number) may always be placed in 2^n ways without
attacking. On an ordinary chessboard n would be 8; therefore 14
bishops may be placed in 256 different ways. It is rather curious that
the general result should come out in so simple a form.
Informational
