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Amusements in Mathematics

improvement was introduced merely to help the eye in actual play. The utility of the
chequers is unquestionable. For example, it facilitates the operation of the bishops,
enabling us to see at the merest glance that our king or pawns on black squares are not
open to attack from an opponent's bishop running on the white diagonals. Yet the
chequering of the board is not essential to the game of chess. Also, when we are
propounding puzzles on the chessboard, it is often well to remember that additional
interest may result from "generalizing" for boards containing any number of squares, or
from limiting ourselves to some particular chequered arrangement, not necessarily a
square. We will give a few puzzles dealing with chequered boards in this general way.
288.—CHEQUERED BOARD DIVISIONS.
I recently asked myself the question: In how many different ways may a chessboard be
divided into two parts of the same size and shape by cuts along the lines dividing the
squares? The problem soon proved to be both fascinating and bristling with difficulties. I
present it in a simplified form, taking a board of smaller dimensions.
It is obvious that a board of four squares can only be so divided in one way—by a straight
cut down the centre—because we shall not count reversals and reflections as different. In
the case of a board of sixteen squares—four by four—there are just six different ways. I
have given all these in the diagram, and the reader will not find any others. Now, take the
larger board of thirty-six squares, and try to discover in how many ways it may be cut
into two parts of the same size and shape.
289.—LIONS AND CROWNS.
The young lady in the illustration is confronted with a little cutting-out difficulty in which
the reader may be glad to assist her. She wishes, for some reason that she has not
communicated to me, to cut that square piece of valuable material into four parts, all of
exactly the same size and shape, but it is important that every piece shall contain a lion
and a crown. As she insists that the cuts can only be made along the lines dividing the
squares, she is considerably perplexed to find out how it is to be done. Can you show her
the way? There is only one possible method of cutting the stuff.
290.—BOARDS WITH AN ODD NUMBER OF SQUARES.
We will here consider the question of those boards that contain an odd number of
squares. We will suppose that the central square is first cut out, so as to leave an even
number of squares for division. Now, it is obvious that a square three by three can only
be divided in one way, as shown in Fig. 1. It will be seen that the pieces A and B are of
the same size and shape, and that any other way of cutting would only produce the same
shaped pieces, so remember that these variations are not counted as different ways. The
puzzle I propose is to cut the board five by five (Fig. 2) into two pieces of the same size
and shape in as many different ways as possible. I have shown in the illustration one way
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