A chessboard is essentially a square plane divided into sixty-four smaller squares by straight lines at right angles.  Originally it was not chequered (that is, made with its rows and columns alternately black and white, or of any other two colours), and this 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.

[Illustration: 

+—–­+—–­*—–­+—–­+   +—–­+—–­+—–­*—–­+   +—–­+—–­+—–­*—–­+
|   |   H   |   |   |   |   |   H   |   |   |   |   H   |
+—–­+—–­*—–­+—–­+   +—–­+—–­*===*—–­+   +—–­*===*—–­*—–­+
|   |   H   |   |   |   |   H   |   |   |   H   H   H   |
+—–­+—–­*—–­+—–­+   +—–­+—–­*—–­+—–­+   +—–­*—–­*—–­*—–­+
|   |   H   |   |   |   |   H   |   |   |   H   H   H   |
+—–­+—–­*—–­+—–­+   +—–­*===*—–­+—–­+   +—–­*—–­*===*—–­+
|   |   H   |   |   |   H   |   |   |   |   H   |   |   |
+—–­+—–­*—–­+—–­+   +—–­*—–­+—–­+—–­+   +—–­*—–­+—–­+—–­+

+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+

+—–­+—–­*—–­+—–­+   +—–­+—–­+—–­*—–­+   +—–­+—–­+—–­*—–­+
|   |   H   |   |   |   |   |   H   |   |   |   |   H   |
+—–­*===*—–­+—–­+   +—–­*===*===*—–­+   +—–­+—–­*===*—–­+
|   H   |   |   |   |   H   |   |   |   |   |   H   |   |
+—–­*===*===*—–­+   +—–­*===*===*—–­+   +—–­+—–­*—–­+—–­+
|   |   |   H   |   |   |   |   H   |   |   |   H   |   |
+—–­+—–­*===*—–­+   +—–­*===*===*—–­+   +—–­*===*—–­+—–­+
|   |   H   |   |   |   H   |   |   |   |   H   |   |   |
+—–­+—–­*—–­+—–­+   +—–­*—–­+—–­+—–­+   +—–­*—–­+—–­+—–­+

]

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.