A PROBLEM IN MOSAICS.
(
Chessboard Problems)
The art of producing pictures or designs by means of joining together
pieces of hard substances, either naturally or artificially coloured, is
of very great antiquity. It was certainly known in the time of the
Pharaohs, and we find a reference in the Book of Esther to "a pavement
of red, and blue, and white, and black marble." Some of this ancient
work that has come down to us, especially some of the Roman mosaics,
would seem to show clearly, even where design is not at first evident,
that much thought was bestowed upon apparently disorderly arrangements.
Where, for example, the work has been produced with a very limited
number of colours, there are evidences of great ingenuity in preventing
the same tints coming in close proximity. Lady readers who are familiar
with the construction of patchwork quilts will know how desirable it is
sometimes, when they are limited in the choice of material, to prevent
pieces of the same stuff coming too near together. Now, this puzzle will
apply equally to patchwork quilts or tesselated pavements.
It will be seen from the diagram how a square piece of flooring may be
paved with sixty-two square tiles of the eight colours violet, red,
yellow, green, orange, purple, white, and blue (indicated by the initial
letters), so that no tile is in line with a similarly coloured tile,
vertically, horizontally, or diagonally. Sixty-four such tiles could not
possibly be placed under these conditions, but the two shaded squares
happen to be occupied by iron ventilators.
[Illustration:
+---+---+---+---+---+---+---+---+
| V | R | Y | G | O | P | W | B |
+---+---+---+---+---+---+---+---+
| W | B | O | P | Y | G | V | R |
+---+---===---+---===---+---+
| G | P H W H V | B H R H Y | O |
+---+---===---+---===---+---+
| R | Y | B | O | G | V | P | W |
+---+---+---+---+---+---+---+---+
| B | G | R | Y | P | W | O | V |
+---+---+---+---+---+---+---+---+
| O | V | P | W | R | Y | B | G |
+---+---+---+---+---+---+---+---+
| P | W | G | B | V | O | R | Y |
+---+---+---+---+---+---+---+---+
|///| O | V | R | W | B | G |///|
+---+---+---+---+---+---+---+---+
]
The puzzle is this. These two ventilators have to be removed to the
positions indicated by the darkly bordered tiles, and two tiles placed
in those bottom corner squares. Can you readjust the thirty-two tiles so
that no two of the same colour shall still be in line?
Answer:
The diagram shows how the tiles may be rearranged. As before, one yellow
and one purple tile are dispensed with. I will here point out that in
the previous arrangement the yellow and purple tiles in the seventh row
might have changed places, but no other arrangement was possible.
