## CHEQUERED BOARD DIVISIONS.

(

Chessboard Problems)

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.

Read Answer

Next:

LIONS AND CROWNS.