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 sub- tracting square is derived from the adding square by reversing the diagonals and exchang- ing 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.
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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.
MAGIC SQUARES OF PRIMES.
The problem of constructing magic squares with prime numbers only was first discussed by myself in The Weekly Dispatch for 22nd July and 5th August 1900; but during the last three or four years it has received great attention from American mathematicians. First, they have sought to form these squares with the lowest possible constants. Thus, the first nine prime numbers, 1 to 23 inclusive, sum to 99, which (being divisible by 3) is theoretically a suitable series; yet it has been demonstrated that the lowest possible constant is in, and the required series as follows : 1, 7, 13, 31, 37, 43, 61, 67, and 73. Similarly, in the case of the fourth order, the lowest series of primes that are "theoretically suitable" will not serve. But in every other order, up to the 12th inclusive, magic squares have been constructed with the lowest series of primes theoretically possible. And the 12th is the lowest order in which a straight series of prime numbers, unbroken, from 1 upwards has been made to work. In other words, the first 144 odd prime numbers have actually been arranged in magic form. The following summary is taken from The Monist (Chicago) for October 1913:—
| Order of Square. | Totals of Series. | Lowest Constants. | Squares made by — |
|---|---|---|---|
| 3rd | 333 | Henry E. Dudeney (1900). | |
| 4th | 408 | 102 | Ernest Bergholt and C. D. Shuldham. |
| 5th | 1065 | 213 | H. A. Sayles |
| 6th | 2448 | 408 | C. D. Shuldham and J N. Muncey. |
| 7th | 4893 | 699 | (C. D. Shuldham and J N. Munceydo. |
| 8th | 8912 | 1114 | (C. D. Shuldham and J N. Munceydo. |
| 9th | 15129 | 1681 | (C. D. Shuldham and J N. Munceydo. |
| 10th | 24160 | 2416 | J. N. Muncey |
| 11th | 36095 | 3355 | J. N. Munceydo. |
| 12th | 54168 | 4514 | J. N. Munceydo. |
For further details the reader should consult the article itself, by W. S. Andrews and H. A. Sayles.
