Although the adding magic square is of such great antiquity, curiously
enough the multiplying magic does not appear to have been mentioned
until the end of the eighteenth century, when it was referred to
slightly by one writer and then forgotten until I revived it in
_Tit-Bits_ in 1897. The dividing magic was apparently first discussed by
me in _The Weekly Dispatch_ in June 1898. The subtracting magic is here
introduced for the first time. It will now be convenient to deal with
all four kinds of magic squares together.
[Illustration: ADDING SUBTRACTING MULTIPLYING DIVIDING]
In these four diagrams we have examples in the third order of adding,
subtracting, multiplying, and dividing squares. In the first the
constant, 15, is obtained by the addition of the rows, columns, and two
diagonals. In the second case you get the constant, 5, by subtracting
the first number in a line from the second, and the result from the
third. You can, of course, perform the operation in either direction;
but, in order to avoid negative numbers, it is more convenient simply to
deduct the middle number from the sum of the two extreme numbers. This
is, in effect, the same thing. It will be seen that the constant of the
adding square is n times that of the subtracting square derived from
it, where n is the number of cells in the side of square. And the
manner of derivation here is simply to reverse the two diagonals. Both
squares are "associated"--a term I have explained in the introductory
article to this department.
The third square is a multiplying magic. The constant, 216, is obtained
by multiplying together the three numbers in any line. It is
"associated" by multiplication, instead of by addition. It is here
necessary to remark that in an adding square it is not essential that
the nine numbers should be consecutive. Write down any nine numbers in
this way--
1 3 5
4 6 8
7 9 11
so that the horizontal differences are all alike and the vertical
differences also alike (here 2 and 3), and these numbers will form an
adding magic square. By making the differences 1 and 3 we, of course,
get consecutive numbers--a particular case, and nothing more. Now, in
the case of the multiplying square we must take these numbers in
geometrical instead of arithmetical progression, thus--
1 3 9
2 6 18
4 12 36
Here each successive number in the rows is multiplied by 3, and in the
columns by 2. Had we multiplied by 2 and 8 we should get the regular
geometrical progression, 1, 2, 4, 8, 16, 32, 64, 128, and 256, but I
wish to avoid high numbers. The numbers are arranged in the square in
the same order as in the adding square.
The fourth diagram is a dividing magic square. The constant 6 is here
obtained by dividing the second number in a line by the first (in either
direction) and the third number by the quotient. But, again, the process
is simplified by dividing the product of the two extreme numbers by the
middle number. This square is also "associated" by multiplication. It is
derived from the multiplying square by merely reversing the diagonals,
and the constant of the multiplying square is the cube of that of the
dividing square derived from it.
The next set of diagrams shows the solutions for the fifth order of
square. They are all "associated" in the same way as before. The
subtracting square is derived from the adding square by reversing the
diagonals and exchanging opposite numbers in the centres of the borders,
and the constant of one is again n times that of the other. The
dividing square is derived from the multiplying square in the same way,
and the constant of the latter is the 5th power (that is the nth) of
that of the former.
[Illustration]
These squares are thus quite easy for odd orders. But the reader will
probably find some difficulty over the even orders, concerning which I
will leave him to make his own researches, merely propounding two little
problems.