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Amusements in Mathematics

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.
The puzzle is to find in how many different ways the four lions may be placed so that
there shall never be more than one lion in any row or column. Mere reversals and
reflections will not count as different. Thus, regarding the example given, if we place the
lions in the other diagonal, it will be considered the same arrangement. For if you hold
the second arrangement in front of a mirror or give it a quarter turn, you merely get the
first arrangement. It is a simple little puzzle, but requires a certain amount of careful
Place as few bishops as possible on an ordinary chessboard so that every square of the
board shall be either occupied or attacked. It will be seen that the rook has more scope
than the bishop: for wherever you place the former, it will always attack fourteen other
squares; whereas the latter will attack seven, nine, eleven, or thirteen squares, according
to the position of the diagonal on which it is placed. And it is well here to state that when
we speak of "diagonals" in connection with the chessboard, we do not limit ourselves to
the two long diagonals from corner to corner, but include all the shorter lines that are
parallel to these. To prevent misunderstanding on future occasions, it will be well for the
reader to note carefully this fact.
Now, how many bishops are necessary in order that every square shall be either occupied
or attacked, and every bishop guarded by another bishop? And how may they be placed?
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