I first deal with a few questions affecting the chessboard itself; then with certain statical puzzles relating to the Rook, the Bishop, the Queen, and the Knight in turn; then dynamical puzzles with the pieces in the same order; and, finally, with some miscellaneous puzzles on the chessboard. It is hoped that the formulae and tables given at the end of the statical puzzles will be of interest, as they are, for the most part, published for the first time. THE CHESSBOARD. "Good company's a chessboard." BYRON'S _Don Juan_, xiii. 89. 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 | | | +---+---*---+---+ +---*---+---+---+ +---*---+---+---+