Magic Squares

Magic Squares are sets of different numbers, each column of which, whether horizontal, vertical, or diagonal, adds alike.

Fig. 1.

Fig. 2.

The above are two examples of a magic square with the same summation (in either case 1891). Considering the difficulty with which a person without previous knowledge could make even one such square, it may surprise many to hear that there are more than 700,000,000,000,000 (seven hundred billion) magic squares of this root (4), with the summation of 1891, each composed of different numbers, or with a different arrangement of the same numbers. Fig. 1 is so constructed that a great variety of other squares may be made from it by altering the four highest numbers in it. Thus, if 13, 14, 15, and 16 be substituted for 1870, 1871, &c. respectively, we get the smallest 4-square possible, with the summation 34. It was at one time thought that magic squares could only be composed of arithmetical or other symmetrical series of numbers; but an examination of Fig. 2 shows that that idea was erroneous.

Within the compass of a short article it is impossible to describe adequately any of the many rules for making magic squares. The following figures will, however, give some idea of the most important method, that of superposition, invented by De la Hire. It is most readily applied to odd squares, more especially to those whose roots are prime numbers. We therefore take the 5-square for our example.

Fig. 3.

Fig. 4.

Each row of the above squares contains the same numbers and in the same order relatively to one another. But in fig. 3 the first number of each row is the same as the third of the row above, whilst in fig. 4 it is the fourth. If now these two squares be combined by adding together the numbers that are in corresponding cells, the resulting square will be magic. In this case it will have the summation of 65, and the top row will be 6, 3, 20, 12, 24. By altering the positions of the numbers in the top rows, and making corresponding alterations in the others, 3600 distinct varieties of this magic square may be obtained.

Although numerous persons have written on magic squares (among whom may be mentioned Leibnitz, Frenicle, De Morgan, Bachet, Ozanam, Montucla, Frost, and Cram), the literature on the subject is by no means easily accessible. Perhaps the best known work is Hutton's Mathematical Recreations; and in this will be found descriptions of other kinds of magic squares, such as the Bordered and Tessellated, which may briefly be described as magic squares within magic squares. Nasik magic squares (so named by Frost from his place of residence in India) are squares whose magicness is not destroyed by repeatedly removing the first column or row to the last place, or vice versa. All squares with prime roots, made by De la Hire's method of superposition, are nasik. Even squares can also be made nasik. Fig. 1, with the numbers 13, 14, 15, and 16 substituted for the four highest, makes a nasik square.