Magic squares have been into fig ures of three dimensions termed magic cubes Diagram N is a magic cube of the fourth order, 1, 2, 3 and 4 are horizontal sections of !N numbered from the top down. There are 52 ranks of numbers in this cube which sum up 130, namely, 16 vertical columns, 16 horizontal rows from front to back, 16 horizontal rows from left to right, and four diagonal lines unit ing four pairs of opposite corners. The sum of any two numbers which are diametrically opposite each other and equidistant from the 0 centre of the cube equals 65; and the sum of the numbers in the 48 sub-squares of four cells each is 130.
Among curiosities in magic construction may be mentioned the following: The square in diagram 0 is filled with the natural numbers in the path of a knight returning to its starting cell, and possesses the property that the differ ence of any two numbers equidistant from and on opposite sides of the centre is 18. It is due to Euler, the famous mathematician.
W. S. Andrews in his Magic Squares and Cubes gives a magic cross filled with 145 numbers, with the statement that it contains the almost incredible number of 160,144 different columns of 21 numbers whose sum is 1,471.
A certain class of magic squares has re ceived much attention in recent years. They are called Nasik squares in England, and in France diabolic squares. They are formed so that the sums along certain lines, such as all the rows, columns, diagonals and broken di agonals are the same. Diagram P is a unique example, as it is composed entirely of prime numbers. It is due to the ingenuity of C. D.
Schuldham and appeared in a recent number of The New York Sun.
This square possesses the Nasik properties above mentioned, and in addition the sum of any two numbers equidistant from the centre and opposite each other is 1,402, or twice the central number.
Magic rectangles, crosses, stars, cylinders, etc., have been constructed, but want of space forbids any further notice of them.
Varying estimates as to the possible number of magic squares of a given order have been made by different investigators. W. W. Rouse Ball in his Mathematical Recreations thinks that those of the fifth order probably exceed half a million. Theodor Hugel in his Die Magischen Quadrate has calculated that the paper required to contain all the magic squares of the 13th order would cover the whole sur face of the earth about 348 times.
As to the scientific value of magic squares the following paragraph is quoted from a paper by Maj. P. A. McMahon, F.RS., published in Proceedings of the Royal Institution of Great Britain, 1892. "What was at first merely a practice of magicians and talisman makers has. now for a long time become' the serious study of mathematicians. . . . It was con sidered possible that some new properties of numbers might be discovered. . . . This has in fact proved to be the case, for from a cer tain point of view the subject has been found to be intimately connected with great depart ments of science such as the Infinitesimal Cal culus, the Calculus of Operations, and the The ory of Groups.