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.





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