(related to Problem: Chequered Board Divisions)
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 center. 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:— $$\begin{array}{ccrcrcrcr} &&a & &b& & c& &\text{Ways}\\ A& =& 8& +& 17& +& 21& =& 46\\ B &=& 0& +& 17& +& 21& =& 38\\ C &=& 15& +& 31& +& 39& =& 85\\ D& =& 17& +& 29& +& 39& =& 85\\ E &=& 1& +& 0& +& 0& = &1\\ \hline &&41&& 94&& 120&& 255 \end{array}$$
I have not attempted the task of enumerating the ways of dividing a board $8\times 8$ — that is, an ordinary chessboard. Whatever the method adopted, the solution would entail considerable labor.
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