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Amusements in Mathematics

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 111, 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
111
Dudeney (1900).
E.
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
do.
8th
8912
1114
do.
9th
15129
1681
do.
10th
24160
2416
J. N. Muncey.
Henry
 
 
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