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(Chessboard Problems)
It will be seen that every square of the board is either occupied or
attacked. The puzzle is to substitute a bishop for the rook on the same
square, and then place the four queens on other squares so that every
square shall again be either occupied or attacked.


[Illustration: FIG. 1.]
[Illustration: FIG. 2.]
The bishop is on the square originally occupied by the rook, and the
four queens are so placed that every square is either occupied or
attacked by a piece. (Fig. 1.)
I pointed out in 1899 that if four queens are placed as shown in the
diagram (Fig. 2), then the fifth queen may be placed on any one of the
twelve squares marked a, b, c, d, and e; or a rook on the two squares,
c; or a bishop on the eight squares, a, b, and e; or a pawn on the
square b; or a king on the four squares, b, c, and e. The only known
arrangement for four queens and a knight is that given by Mr. J. Wallis
in _The Strand Magazine_ for August 1908, here reproduced. (Fig. 3.)
[Illustration: FIG. 3.]
I have recorded a large number of solutions with four queens and a rook,
or bishop, but the only arrangement, I believe, with three queens and
two rooks in which all the pieces are guarded is that of which I give an
illustration (Fig. 4), first published by Dr. C. Planck. But I have
since found the accompanying solution with three queens, a rook, and a
bishop, though the pieces do not protect one another. (Fig. 5.)
[Illustration: FIG. 4.]
[Illustration: FIG. 5.]

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