Magic Squares and Cubes

Jan 27, 2013 by

Magic Squares and Cubes by W. S. Andrews was originally published in 1917 and describes magic squares, cubes, spheres, and other shapes. As well as their properties, and how to make them.

A few lines from the introduction to give you a feel for this book:

“The peculiar interest of magic squares in general lies in the fact that they possess the charm of mystery. They appear to betray some hidden intelligence which by a preconceived plan produces the impression of intentional design, a phenomenon which finds its close analogue in nature. Although magic squares have no immediate practical use, they have always exercised a great influence upon thinking people. It seems to me that they contain a lesson of great value in being a palpable instance of the symmetry of mathematics, throwing thereby a clear light upon the order that pervades the universe wherever we turn, in the inflnitesimally small interrelations of atoms as well as in the immeasurable domain of the starry heavens, an order which, although of a different kind and still more intricate, is also traceable in the development of organized life, and even in the complex domain of human action.

Pythagoras says that number is the origin of all things, and certainly the law of number is the key that unlocks the secrets of the universe. But the law of number possesses an immanent order, which is at first sight mystifying, but on a more intimate acquaintance we easily understand it to be intrinsically necessary ; and this law of number explains the wondrous consistency of the laws of nature. Magic squares are conspicuous instances of the intrinsic harmony of number, and so they will serve as an interpreter of the cosmic order that dominates all existence. Though they are a mere intellectual play they not only illustrate the nature of mathematics, but also, incidentally, the nature of existence dominated by mathematical regularity.

In arithmetic we create a universe of figures by the process of counting ; in geometry we create another universe by drawing lines in the abstract field of imagination, laying down definite directions ; in algebra we produce magnitudes of a still more abstract nature, expressed by letters. In all these cases the first step producing the general conditions in which we move, lays down the rule to which all further steps are subject, and so every one of these universes is dominated by a consistency, producing a wonderful symmetry. There is no science that teaches the harmonies of nature more clearly than mathematics, and the magic squares are like a mirror which reflects the symmetry of the divine norm immanent in all things, in the immeasurable immensity of the cosmos and in the construction of the atom not less than in the mysterious depths of the human mind. Paul Carus’ study of magic squares probably dates back to prehistoric¬† times. Examples have been found in Chinese literature written about A. D. 1 125 which were evidently copied from still older documents.

It is recorded that as early as the ninth century magic squares were used by Arabian astrologers in their calculations of horoscopes etc. Hence the probable origin of the term “magic” which has survived to the present day. A magic square consists of a series of numbers so arranged in a square, that the sum of each row and column and of both the corner diagonals shall be the same amount which may be termed the summation (S). Any square arrangement of numbers that fulfills these conditions may properly be called a magic square. Various features may be added to such a square which may enhance its value as a mathematical curio, but these must be considered non-essentials. There are thus many different kinds of magic squares, but this chapter will be devoted principally to the description of associated or regular magic squares, in which the sum of any two numbers that are located in cells diametrically equidistant from the center of the square equals the sum of the first and last terms of the series, or n 2 -j- 1.

Magic squares with an odd number of cells are usually constructed by methods which differ from those governing the construction of squares having an even number of cells, so these two classes will be considered under separate headings. The square of 3 X 3 shown in Fig. 1 covers the smallest aggregation of numbers that is capable of magic square arrangement, and it is also the only possible arrangement of nine different numbers, relatively to each other, which fulfills the required conditions. It will be seen that the sum of each of the three vertical, the three horizontal, and the two corner diagonal columns in this square is 15, making in all eight columns having that total: also that the sum of any two opposite numbers is 10, which is twice the center number, or n 2 -f- 1. ”

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