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.
Answer:
There are 255 different ways of cutting the board into two pieces of
exactly the same size and shape. Every way must involve one of the five
cuts shown in Diagrams A, B, C, D, and E. To avoid repetitions by
reversal and reflection, we need only consider cuts that enter at the
points a, b, and c. But the exit must always be at a point in a straight
line from the entry through the centre. This is the most important
condition to remember. In case B you cannot enter at a, or you will get
the cut provided for in E. Similarly in C or D, you must not enter the
key-line in the same direction as itself, or you will get A or B. If you
are working on A or C and entering at a, you must consider joins at one
end only of the key-line, or you will get repetitions. In other cases
you must consider joins at both ends of the key; but after leaving a in
case D, turn always either to right or left--use one direction only.
Figs. 1 and 2 are examples under A; 3 and 4 are examples under B; 5 and
6 come under C;
and 7 is a pretty example of D. Of course, E is a peculiar type, and
obviously admits of only one way of cutting, for you clearly cannot
enter at b or c.
Here is a table of the results:--
a b c Ways.
A = 8 + 17 + 21 = 46
B = 0 + 17 + 21 = 38
C = 15 + 31 + 39 = 85
D = 17 + 29 + 39 = 85
E = 1 + 0 + 0 = 1
-- -- -- ---
41 94 120 255
I have not attempted the task of enumerating the ways of dividing a
board 8 x 8--that is, an ordinary chessboard. Whatever the method
adopted, the solution would entail considerable labour.
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