Easy Methods of Constructing the Various Types of Magic Squares and Magic Cubes, with Symmetric Designs founded Thereon

Abstract

A MAGIC square is an example of a problem which is a particular case of another which from its enunciation may be subjected to mathematical analysis. The n² cells of a square of order n may be supposed occupied, each of them, by one or more numbers in such wise that the sums of the numbers in the n rows and in the n columns have given values varying from row to row and from column to column. The enumeration of such squares, or more generally of such rectangles, has been made the subject of mathematical investigation employing algebraic symmetric functions and the allied differential operators, and complete success has resulted. The absolute magnitude of the numbers appearing in the cells may be restricted, and any number of the cells may be empty; no additional difficulties present themselves. Other problems of enumeration of the magic square kind, of which the simplest is known as the Latin square, first examined by Euler, and by others up to the time of Cayley, have also in recent years completely yielded to the same calculus of symmetric functions. In all these cases row and column properties are dealt with, but directly we introduce what may be termed diagonal properties the analysis fails to overcome the great difficulties which are thereby imported into the problems. The problem of the magic square involves restrictions and limitations of the general problem above mentioned which render it impossible of complete treatment by any known method of analysis. The condition that the first n² natural numbers must appear one, and only one, in each cell, is, to begin with, of a most difficult character, and is made more so by the importation of the diagonal conditions. Many kinds of magic squares, moreover, involve additional properties connected with broken diagonals, nuclear squares, symmetrically placed cells, &c., which mathematically are of a most arduous nature. It thence arises that though many mathematicians of repute from the earliest times have studied the subject, they have devoted their labours mainly to observational and tentative methods of actually constructing different classes of these squares, and have seldom seriously attempted the enumeration.

Easy Methods of Constructing the Various Types of Magic Squares and Magic Cubes, with Symmetric Designs founded Thereon.

By Dr. John Willis. Pp. 256. (Bradford and London: Percy Lund, Humphries and Co., Ltd., 1909.)