[Illustration]




SUBTRACTING, MULTIPLYING, AND DIVIDING MAGICS.

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