A Magic Square is a series of figures arranged in the equal divisions
of a square in such a manner that the figures in each row when added
up, whether horizontally, vertically, or diagonally, form exactly the
same sum.
They have been called “Magic” because the ancients ascribed to them
great virtues, and because this arrangement of numbers formed the
basis and principle of their talismans. Archimedes devoted a great
amount of attention to them, which has caused a great many to speak
of them as “the squares of Archimedes.” They may be either odd or
even. When the former, the following method will be found valuable:--
With the digits from 1 to 25 form a square so that the numbers when
added up horizontally, vertically, or diagonally will amount to 65.
_Method._--Imagine an exterior line of squares above the magic square
you wish to form, and another on the right hand of it. These two
imaginary lines are shown in the diagram.
18 25 2 9
+----+----+----+----+----+
| 17 | 24 | 1 | 8 | 15 | 17
+----+----+----+----+----+
| 23 | 5 | 7 | 14 | 16 | 23
+----+----+----+----+----+
| 4 | 6 | 13 | 20 | 22 | 4
+----+----+----+----+----+
| 10 | 12 | 19 | 21 | 3 | 10
+----+----+----+----+----+
| 11 | 18 | 25 | 2 | 9 |
+----+----+----+----+----+
1st. In placing the numbers in the square, we must go in the
ascending diagonal direction from left to right, any number which, by
pursuing this direction, would fall into the exterior line must be
carried along that line of squares, whether vertical or horizontal,
to the last square. Thus, 1 having been placed in the centre of the
top row, 2 would fall into the exterior square above the fourth
vertical line; then ascending diagonally 3 falls into the square
diagonally from 2, but 4 falls out of it to the end of a horizontal
line, and it must be carried along that line to the extreme left and
there placed. Resuming our diagonal ascension to the right we place 5
where the reader sees it, and would place 6 in the middle of the top
row, but as we find 1 is already there we look for the direction to
2nd. That when in ascending diagonally we come to a square already
occupied, we must place the number which, according to the 1st rule
should go into that occupied square directly under the last number
placed: thus, in ascending with 4, 5, 6, the 6 must be placed under
the 5, because the square next to 5 in diagonal direction is occupied.
A Promising Sign--I O U.