“Magic Squares” Were Held in Veneration by the Egyptians and
Pythagoreans, and They Constitute the Oldest Numerical Problems Known to
Man—Bewildering Results Obtained by Simple Methods.
The art of arranging numbers in the form of squares, so that the sum of
the various rows—vertical, horizontal, and diagonal—would in each case
be the same, is, without question, the oldest of mathematical puzzles.
The Egyptians and Pythagoreans held them in the greatest
veneration—especially the latter, who dedicated them to the then known
seven planets.
The magic 34 square was probably the strangest freak of figures known at
this time.
┌──┬──┬──┬──┐
│16│ 3│ 2│13│
├──┼──┼──┼──┤
│ 5│10│11│ 8│
├──┼──┼──┼──┤
│ 9│ 6│ 7│12│
├──┼──┼──┼──┤
│ 4│15│14│ 1│
└──┴──┴──┴──┘
This strange freak may be found in Dürer’s “Melancholia,” engraved on
copper in 1514, being included in the series of symbolical engravings
of “The Death of the Devil,” “The Knight on Horseback,” etc.
The aim in this instance, as shown by ancient writings, was not only
to obtain the same total (34) in the ten rows of four, but to discover
as many symmetrical combinations as possible giving the same result.
According to the ancients, “symmetrical combinations which no man
could number” were to be found in this arrangement of the numbers from
1 to 16, inclusive. As an example, take 16, 3, 5, and 10, or 2, 8, 9,
and 15, or 1, 9, 16, and 8, and so on indefinitely. The result is the
same.
Another unique example is the following:
┌──┬──┬──┬──┬──┐
│ 3│20│ 7│24│11│
├──┼──┼──┼──┼──┤
│16│ 8│25│12│ 4│
├──┼──┼──┼──┼──┤
│ 9│21│13│ 5│17│
├──┼──┼──┼──┼──┤
│22│14│ 1│18│10│
├──┼──┼──┼──┼──┤
│15│ 2│19│ 6│23│
└──┴──┴──┴──┴──┘
In this case the sum is 65, and can be reached in an almost endless
variety of combinations. However, there is one feature to be
remembered in dealing with this problem, and that is that the central
number (13) must be added to each combination except in the straight
and diagonal lines. Thus: 20, 24, 2, 6, and 13, or 8, 12, 14, 18, and
13, etc., each make the magic sum 65.
The well-known “15 puzzle” is another illustration of the surprising
feats which figures are sometimes made to play. The problem being to
arrange in a square of three rows, three figures in each row, the
numerals, 1 to 9 inclusive, in such a manner that each row—vertical,
horizontal, or diagonal—will total 15. This is more difficult than
appears at first glance, unless you have the key, which is: place 5 in
the center, and let the four corners be 2, 4, 6, and 8. The rest is
easy.
┌──┬──┬──┐
│ 2│ 9│ 4│
├──┼──┼──┤
│ 7│ 5│ 3│
├──┼──┼──┤
│ 6│ 1│ 8│
└──┴──┴──┘
This form differs from the 65 and 34 in that it can only be added
diagonally, horizontally, and vertically.
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