Magic Square

MAGIC SQUARE, an arrangement of numbers in the form of a square so that every column, every row, and each of the two diagonals, add up alike, this sum being called the constant. Strictly speaking the numbers used should be consecutive from I up to where n is the number of cells (or squares) in any side. Such an arrangement of n' numbers is called a square of the nth order. Thus fig. I complies with all the conditions and is a magic square of the 3rd order. These squares are of very great antiquity and appear to have been known from very ancient times in China and India, where, as at the present time, magic squares were worn engraved on metal or stone as amulets or talismans.

They appear to have been introduced into Europe early in the Christian era though the first known writer on the subject was Emanuel Moschopoulus, a Greek of uncertain date who lived in Constantinople, probably about 1300. His work in manuscript is in the National Library in Paris (ms. No. 2,428). Cornelius Agrippa (1486-1535) constructed squares of the orders 3, 4, 5, 6, 7, 8, 9, which were associated with the seven astrological fig. 6, which is a perfect magic square. Similar methods have been devised for the construction of magic squares and some of these will be found in the books named at the end of this article. But most of the writers on the subject develope their own favourite schemes. It is not difficult to construct squares of particular types but the ideal solution is, of course, a method completely general for squares of every order, that will include every possible ar rangement of the order dealt with. Such a solution is probably not discoverable. Magic squares of singly-even orders (such as those where n=6, io, 14, 18, etc.) are generally the most difficult of all to construct and their treatment is largely empirical. Large "planets," Saturn, Jupiter, Mars, the Sun, Venus, Mercury and the Moon. The magic square of the 4th order shown in fig. 2 is to be seen in Albrecht Durer's picture of "Melancholy." The date of the work (1514) is indicated in the two central cells of the bottom row, but whether this was intentional or a mere coinci dence is not known (see Notes and Queries, Feb.-March 1918). In later times the subject has been investigated as a mathematical curiosity and has a large though scattered literature of its own. The three main lines of enquiry are construction, enumeration and classification.

Construction.-It

is convenient to deal with these squares in three classes. Those of an odd order, those of a doubly-even order (that is where it is of the form 4m) and those of a singly even order (where n is of the form 2 [2111+ . These vary in difficulty in their order. The smallest possible square of an odd order is, of course, that of the 3rd order shown in fig. 1, to which there is that one fundamental solution only. Eight different ar rangements may be obtained by merely turning the page round and also turning it round in front of a mirror. These so-called rever sals and reflections are not usually counted as different. A square of the odd order may immediately be written down in the man ner shown in fig. 3, described first by De la Loubere. Here we start at the central cell in the top row and proceed diagonally upwards to the right. As the 2 is outside the square we bring it to the bottom of the column, thus giving it the position it occupies on the outside square. Having written the 3, the 4 falls outside the square, so we insert it at the opposite end of the row and write in the 5. As the next square is occupied by the 1 we write the 6 beneath the 5 and proceed until the 1 o falls outside the square and so on. It will be noted that 16 falling outside the square at a corner is written be neath the 15 as in the case of an occupied square. This can be ap plied to any square of an odd order, but, like so many methods that have been adopted, it only produces one type of square, though it may be modified to form a limited number of other squares. Thus we may start with the 1 in any cell and always get a square that is magic in rows and columns, but not necessarily in the diagonals. Bachet (1612) used a somewhat similar method. Another method devised by De la Hire, may be used for the construction of squares of any order. He employed three subsidiary squares though one of these has been dispensed with by later writers. In fig. 4, the numbers 1, 2, 3, 4, 5, are arranged so that every number appears once and once only in every row and column, and in one diagonal, the other diagonal having repetitions of 3. In fig. 5, the numbers o, 5, 15, 20, are similarly treated only the repeated io's are in the other diagonal. These squares superimposed and added produce collections of squares of the higher orders have been compiled by Violle and others, so that examples are easily obtainable.