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Amusements in Mathematics

Magic Square Problems
"By magic numbers."
CONGREVE, The Mourning Bride.
This is a very ancient branch of mathematical puzzledom, and it has an immense, though
scattered, literature of its own. In their simple form of consecutive whole numbers
arranged in a square so that every column, every row, and each of the two long diagonals
shall add up alike, these magic squares offer three main lines of investigation:
Construction, Enumeration, and Classification. Of recent years many ingenious methods
have been devised for the construction of magics, and the law of their formation is so
well understood that all the ancient mystery has evaporated and there is no longer any
difficulty in making squares of any dimensions. Almost the last word has been said on
this subject. The question of the enumeration of all the possible squares of a given order
stands just where it did over two hundred years ago. Everybody knows that there is only
one solution for the third order, three cells by three; and Frénicle published in 1693
diagrams of all the arrangements of the fourth order—880 in number—and his results
have been verified over and over again. I may here refer to the general solution for this
order, for numbers not necessarily consecutive, by E. Bergholt in Nature, May 26, 1910,
as it is of the greatest importance to students of this subject. The enumeration of the
examples of any higher order is a completely unsolved problem.
As to classification, it is largely a matter of individual taste—perhaps an æsthetic
question, for there is beauty in the law and order of numbers. A man once said that he
divided the human race into two great classes: those who take snuff and those who do
not. I am not sure that some of our classifications of magic squares are not almost as
valueless. However, lovers of these things seem somewhat agreed that Nasik magic
squares (so named by Mr. Frost, a student of them, after the town in India where he lived,
and also called Diabolique and Pandiagonal) and Associated magic squares are of special
interest, so I will just explain what these are for the benefit of the novice. I published in
The Queen for January 15, 1910, an article that would enable the reader to write out, if he
so desired, all the 880 magics of the fourth order, and the following is the complete
classification that I gave. The first example is that of a Simple square that fulfils the
simple conditions and no more. The second example is a Semi-Nasik, which has the
additional property that the opposite short diagonals of two cells each together sum to 34.
Thus, 14 + 4 + 11 + 5 = 34 and 12 + 6 + 13 + 3 = 34. The third example is not only Semi-
Nasik but also Associated, because in it every number, if added to the number that is
equidistant, in a straight line, from the centre gives 17. Thus, 1 + 16, 2 + 15, 3 + 14, etc.
The fourth example, considered the most "perfect" of all, is a Nasik. Here all the broken
diagonals sum to 34. Thus, for example, 15 + 14 + 2 + 3, and 10 + 4 + 7 + 13, and 15 + 5
+ 2 + 12. As a consequence, its properties are such that if you repeat the square in all
directions you may mark off a square, 4 × 4, wherever you please, and it will be magic.
The following table not only gives a complete enumeration under the four forms
described, but also a classification under the twelve graphic types indicated in the
diagrams. The dots at the end of each line represent the relative positions of those
complementary pairs, 1 + 16, 2 + 15, etc., which sum to 17. For example, it will be seen
 
 
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