"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.
[Illustration: SIMPLE]
[Illustration: SEMI-NASIK]
[Illustration: ASSOCIATED]
[Illustration: NASIK]
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 that the first
and second magic squares given are of Type VI., that the third square is
of Type III., and that the fourth is of Type I. Edouard Lucas indicated
these types, but he dropped exactly half of them and did not attempt the
classification.
NASIK (Type I.) . . . . . 48
SEMI-NASIK (Type II., Transpositions
of Nasik) . 48
" (Type III., Associated) 48
" (Type IV.) . . . 96
" (Type V.) . . . 96 192
___
" (Type VI.) . . . 96 384
___
SIMPLE. (Type VI.) . . . 208
" (Type VII.) . . . 56
" (Type VIII.). . . 56
" (Type IX.) . . . 56
" (Type X.) . . . 56 224
___
" (Type XI.) . . . 8
" (Type XII.) . . . 8 16 448
___ ___ ___
880
___
It is hardly necessary to say that every one of these squares will
produce seven others by mere reversals and reflections, which we do not
count as different. So that there are 7,040 squares of this order, 880
of which are fundamentally different.
An infinite variety of puzzles may be made introducing new conditions
into the magic square. In _The Canterbury Puzzles_ I have given examples
of such squares with coins, with postage stamps, with cutting-out
conditions, and other tricks. I will now give a few variants involving
further novel conditions.