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.

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.

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 sixty-four 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.