THE FIVE CRESCENTS OF BYZANTIUM.
(
Chessboard Problems)
When Philip of Macedon, the father of Alexander the Great, found himself
confronted with great difficulties in the siege of Byzantium, he set his
men to undermine the walls. His desires, however, miscarried, for no
sooner had the operations been begun than a crescent moon suddenly
appeared in the heavens and discovered his plans to his adversaries. The
Byzantines were naturally elated, and in order to show their gratitude
they erected a statue to Diana, and the crescent became thenceforward a
symbol of the state. In the temple that contained the statue was a
square pavement composed of sixty-four large and costly tiles. These
were all plain, with the exception of five, which bore the symbol of the
crescent. These five were for occult reasons so placed that every tile
should be watched over by (that is, in a straight line, vertically,
horizontally, or diagonally with) at least one of the crescents. The
arrangement adopted by the Byzantine architect was as follows:--
Now, to cover up one of these five crescents was a capital offence, the
death being something very painful and lingering. But on a certain
occasion of festivity it was necessary to lay down on this pavement a
square carpet of the largest dimensions possible, and I have shown in
the illustration by dark shading the largest dimensions that would be
available.
The puzzle is to show how the architect, if he had foreseen this
question of the carpet, might have so arranged his five crescent tiles
in accordance with the required conditions, and yet have allowed for the
largest possible square carpet to be laid down without any one of the
five crescent tiles being covered, or any portion of them.
Answer:
If that ancient architect had arranged his five crescent tiles in the
manner shown in the following diagram, every tile would have been
watched over by, or in a line with, at least one crescent, and space
would have been reserved for a perfectly square carpet equal in area to
exactly half of the pavement. It is a very curious fact that, although
there are two or three solutions allowing a carpet to be laid down
within the conditions so as to cover an area of nearly twenty-nine of
the tiles, this is the only possible solution giving exactly half the
area of the pavement, which is the largest space obtainable.
