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.


407.--TWO NEW MAGIC SQUARES.