MAGIC SQUARE. This term Is applied to set of numbers arranged in a square in such a manner that the vertical, horizontal,and diagonal columns shall give the same sums. Such arrangements were known very early to the Hindus, Egyptians, and Chinese, among whom, as also among the Europeans of the middle ages, a belief existed that such squares bad astrological and divinatory qualities. Emanuel 3Ieschopulta. of Constantinople, wrote on them in Greek in the middle of the 15th century. Others who have written on the subject are Leibnitz, Frrnicle, 13achet, La Hire, Saurin, &c. (See 3fontuela's History.' voL L, p. 346; ' Eneyelm;6die 316th.; article ' Quarr4a =gigues ; ' Hutton'.' Dictionary ; ' and the' Mathematical Recreations' of the same author.) Though the question of magic squares be in itself of no use, yet it belongs to a clams of problems which call into action a beneficial species of investigation. Without laying down any rules for their construction, we shall content ourselves with destroying their magio quality, and ahowin? that the non-existence of such squares would be much more surprising than their existence.
Take any set of numbers in arithmetical progression, and such that their number shall be a square number, say the first sixteen numbers 1 2 3 4 5 6 7 8 16 15 14 13 12 11 10 any of these in the first half, with its corresponding number in the second half, makes up 17. Write the numbers in the following manner : Take four of theme in such a manner as to take one out of each row, and one out of each column, and it will be found, and may easily be proved, that the sum of numbers in every such set must consist of two pairs of corresponding numbers, so that their sum must bo twice 17, or 34. The different ways in which this can be done are in number 4 x 3 x 2 x 1, or 24, as follows. Out of theme subdivisions a set may be taken from each, so that no number shall be repeated, in 24 different ways, as in the following sample, which shows the four ways that begin with 1 6 11 16.
' Now in each of these 24 squares, every horizontal row can be written in 24 orders [Comm:canoes], and in putting the different orders together, each square admits of 24 x 24 x 24 x 24, or 331,776 arrangements, without altering the horizontal rows, but only the order of the figures in each row. But the order of the horizontal rows can be varied 24 ways in each square, and there are 24 squares: so that we have 331.776 x 24 x 24, or 101,102,976 squares, no ouo of which repeats any number more than once, and in every one of which the sum of any horizontal row Is 34, made by two pairs of numbers which give 17 each. But the number of ways of forming 34 out of four of the first sixteen numbers is not yet exhausted : for, taking any one set, say 1 16 11 6 in which 1 and 6 correspond to 16 and 11, we may write 2 and 5, or 3 and 4, for 1 and 6,so that we have not included In the preceding list 2 16 11 5 3 16 11 4 with all their variations of order ; and similar ones for all the rest of the list. It would be almost impossible to doubt that in many of this enormous number of squares, the vertical columns will sometimes be Baas of these new seta : and it would be something short of magic if sane should not also have diagonal columns which fulfil the same condi tion. In fact, Freniele has shown 880 methods of making these squares magical, a few of which are as follows (' Divers Ouvrages, &c. Parisi 1693) : In Freniclo's list of 8S0, only those squares are included which are essentially different : thus the following four, which may be made by turning the last square into different positions, count only as one.
The methods which have been given for the formation of magic squares are divided into different rules, according as the number iu each side is odd, evenly even, or oddly oven. A general method which shall apply to all cases ie yet wanting. For a full account of these rules see Hutton'a Mathematical Recreations.'