
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 TitBits 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.
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—
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—
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.
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.
407.—TWO
NEW MAGIC SQUARES.
Construct a subtracting magic square with the first sixteen whole numbers
that shall be "associated" by subtraction. The constant is, of course,
obtained by subtracting the first number from the second in line, the result
from the third, and the result again from the fourth. Also construct a dividing
magic square of the same order that shall be "associated" by division.
The constant is obtained by dividing the second number in a line by the first,
the third by the quotient, and the fourth by the next quotient.
Solution 408.—MAGIC
SQUARES OF TWO DEGREES.
While reading a French mathematical work I happened to come across, the
following statement: "A very remarkable magic square of 8, in two degrees, has
been constructed by M. Pfeffermann. In other words, he has managed to dispose
the sixtyfour first numbers on the squares of a chessboard in such a way that
the sum of the numbers in every line, every column, and in each of the two
diagonals, shall be the same; and more, that if one substitutes for all the
numbers their squares, the square still remains magic." I at once set to work to
solve this problem, and, although it proved a very hard nut, one was rewarded by
the discovery of some curious and beautiful laws that govern it. The reader may
like to try his hand at the puzzle.
Solution
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