BOARDS WITH AN ODD NUMBER OF SQUARES.
(
Chessboard Problems)
We will here consider the question of those boards that contain an odd
number of squares. We will suppose that the central square is first cut
out, so as to leave an even number of squares for division. Now, it is
obvious that a square three by three can only be divided in one way, as
shown in Fig. 1. It will be seen that the pieces A and B are of the same
size and shape, and that any other way of cutting would only produce the
same shaped pieces, so remember that these variations are not counted as
different ways. The puzzle I propose is to cut the board five by five
(Fig. 2) into two pieces of the same size and shape in as many different
ways as possible. I have shown in the illustration one way of doing it.
How many different ways are there altogether? A piece which when turned
over resembles another piece is not considered to be of a different
shape.
[Illustration:
+------+---+
| H | |
+---===---+
| HHHHH |
+---===---+
| | H |
+---+------+
Fig 1]
[Illustration:
+---+---+---+---+---+
| | | | | |
=========---+---+
| | | H | |
+---+---===---+---+
| | HHHHH | |
+---+---===---+---+
| | H | | |
+---+---=========
| H | | | |
+------+---+---+---+
Fig 2]
Answer:
There are fifteen different ways of cutting the 5 x 5 board (with the
central square removed) into two pieces of the same size and shape.
Limitations of space will not allow me to give diagrams of all these,
but I will enable the reader to draw them all out for himself without
the slightest difficulty. At whatever point on the edge your cut enters,
it must always end at a point on the edge, exactly opposite in a line
through the centre of the square. Thus, if you enter at point 1 (see
Fig. 1) at the top, you must leave at point 1 at the bottom. Now, 1 and
2 are the only two really different points of entry; if we use any
others they will simply produce similar solutions. The directions of the
cuts in the following fifteen
[Illustration: Fig. 1. Fig. 2.]
solutions are indicated by the numbers on the diagram. The duplication
of the numbers can lead to no confusion, since every successive number
is contiguous to the previous one. But whichever direction you take from
the top downwards you must repeat from the bottom upwards, one direction
being an exact reflection of the other.
1, 4, 8.
1, 4, 3, 7, 8.
1, 4, 3, 7, 10, 9.
1, 4, 3, 7, 10, 6, 5, 9.
1, 4, 5, 9.
1, 4, 5, 6, 10, 9.
1, 4, 5, 6, 10, 7, 8.
2, 3, 4, 8.
2, 3, 4, 5, 9.
2, 3, 4, 5, 6, 10, 9.
2, 3, 4, 5, 6, 10, 7, 8.
2, 3, 7, 8.
2, 3, 7, 10, 9.
2, 3, 7, 10, 6, 5, 9.
2, 3, 7, 10, 6, 5, 4, 8.
It will be seen that the fourth direction (1, 4, 3, 7, 10, 6, 5, 9)
produces the solution shown in Fig. 2. The thirteenth produces the
solution given in propounding the puzzle, where the cut entered at the
side instead of at the top. The pieces, however, will be of the same
shape if turned over, which, as it was stated in the conditions, would
not constitute a different solution.
